Integrable deformation of $\mathbb{CP}^n$ and generalised Kaehler geometry

We build on the results of arXiv:1912.11036 for generalised frame fields on generalised quotient spaces and study integrable deformations for $\mathbb{CP}^n$. In particular we show how, when the target space of the Principal Chiral Model is a complex projective space, a two-parameter deformation can, in principle, be introduced. The second parameter can however be removed via a diffeomorphism, which we construct explicitly, in accordance with the results stemming from a thorough integrability analysis we carry out. We also elucidate how the deformed target space can be seen as an instance of generalised Kaehler, or equivalently bi-Hermitian, geometry. In this respect, we find explicit expressions for generalised Kaehler potential and/or pure spinors for $\mathbb{CP}^n$ with $n = 1,2,3$.


Introduction
Integrable (classical) field theories, a prototypical example being the principal chiral model (PCM), offer a particularly tractable -and yet rich -class of theories. Given the principal chiral model on a group manifold G, one might wish to deform it in such a way so as to preserve its integrability whilst generalising the resulting target space geometry, possibly adding a B-field contribution. To this end, two models have been particularly successful, namely the η-deformation of the PCM (sometimes called Yang-Baxter model) and the λ-deformation of WZW CFTs, introduced respectively by Klimcik [2] and Sfetsos [3].
Recent work has highlighted a close connection between such integrable models and Pois son-Lie (PL) symmetry. A non-linear σ-model is said to have a Poisson-Lie symmetry when the target space manifold, M, admits the action of a Lie group G (with algebra g) for which the corresponding world sheet Noether currents are not conserved (i.e. G is not an isometry group) but obey a modified on-shell conservation law that is non-commutative with respect to a "dual" algebrag [4,5]. Such PL models have an elegant Hamiltonian formalism, now called the E-model [6][7][8], describing currents on a Drinfel'd double, i.e. a 2D-dimensional Lie group, D, whose algebra, d, can be decomposed into two maximally isotropic subalgebras g andg with respect to an ad-invariant inner product. The η-deformation of the PCM on G falls into this class [8] with the Drinfel'd double given by the complexification of the Lie algebra, d = g C . Relaxing the requirement that d be a Drinfel'd double allows the E-model to also describe the λ-deformation of WZW CFTs [9,10].
The above construction has been generalised in two key ways. First one can consider a multi-parametric integrable deformation of the theories on group manifolds. As an example, the so-called bi-Yang-Baxter deformation [11,12] augments the deformation parameter, η, of the Yang-Baxter model with a second deformation parameter, ζ, whilst retaining its (classical) integrability. Further work in this direction [13][14][15] extends this model including a Wess-Zumino term.
A second generalisation is to consider models defined on spaces other than group manifolds. This was initiated in [5], where E-models with a gauge symmetry are shown to give rise to the "dressing coset" construction of Poisson-Lie T-duality. This construction is motivated by the observation that a description without a gauge symmetry is not suitable to capture Abelian and non-Abelian T-duality which act on isometries with fixed point. The additional data this construction requires is an isotropic subgroup H which allows one to realise the target space of the underlying non-linear σ-model as the double coset M = H\D/ G. This setting has come under renewed interest since it includes (the bosonic sector of) integrable deformations of the AdS 5 × S 5 superstring [2, [16][17][18][19][20][21][22][23]. Recently, Poisson-Lie T-duality and variations thereof were studied in the context of non-geometric backgrounds [24].
In this article we will consider integrable deformations on a particular class of coset spaces where h = Lie(H) is coisotopic. In a first instance, they result in a very natural Ansatz for a deformed Lagrangian containing two deformation parameters. Although such "double deforma tions" have already been considered in a previous work [25], their integrability and geometric properties have so far remained quite obscure. In fact, these models come with a precise con straint, detailed in Appendix A, which allows for the identification of their target spaces as Poisson homogeneous spaces. In order to address integrability, we shall require these spaces to be additionally Hermitian and symmetric. As it turns out, complex projective spaces CP n are the natural setting where to carry out our investigation. They are exemplars of a broader class of algebraic varieties, known as generalised flag manifolds, which have recently benefited, alongside para-complex manifolds, from a surge of interest in the physics literature as they materialise in rich examples [26][27][28][29][30][31].
The key results of this paper include: subgroup) we demonstrate that the deformed target space geometry is generalised Käh ler [32,33] (equivalently, bi-Hermitian [34]). As an example, and with future holographic applications in mind, we consider the case of G/H= CP n and construct its generalised Kähler structure. We show that for n > 1 the space is parametrised by semi-chiral multi plets plus an additional chiral multiplet when n is odd. If n = 1 a single chiral multiplet turns out to be sufficient. For CP 1 and CP 2 we give explicit coordinate expressions and provide the generalised Kähler potentials that entirely specifies the geometry. For CP 3 we give an explicit and elegant form for the pure spinors corresponding to the generalised complex structures. 3 Writing down a tentative two-parameter deformation when G/H is also a symmetric space we prove, by giving an explicit Lax connection, the classical integrability of this would-be double deformation. We complete the analysis by finding the twist function describing the Maillet algebra [35] of such models. As a byproduct of this analysis, we identify a new tension and deformation parameter making the double deformation equivalent to the single parameter deformation. We confirm this by constructing a diffeomorphism undoing the deformation induced by the second parameter in an appropriate chart.
The outline of the remainder of this paper is as follows: in Section 2 we discuss the world sheet of the theories under consideration taking both Lagrangian non-linear σ-model and Hamil tonian E-model perspectives. In Section 3 we focus our attention to a class of two-parameter integrable Poisson-Lie coset models constructing their Lax representation and demonstrating that they fulfil the requirement of strong integrability. In Section 4 we show that they induce a generalised Kähler structure on the target space, culminating in some explicit examples based on CP n .

World sheet perspective
The generalised coset construction which we will present in the next section was originally motivated by closed string world sheet theories with Poisson-Lie symmetry. We shall begin in Section 2.1 by presenting a conventional Poisson-Lie σ-model, taking the opportunity to review some known features. From this perspective one can readily extract conventional target space geometric data even though the underlying algebraic structure is rather obscured. To expose this structure it is helpful to adopt a Hamiltonian approach called the E-model [8] -introduced in Section 2.2 -and construct the corresponding Poisson brackets.

Poisson-Lie σ-models and subgroup invariance
The Poisson-Lie non-linear σ-model on a Lie group G is defined by the action 1 in which e a ± denote the pull-backs of the left-invariant Maurer-Cartan one-forms e a = e a i dx i onto the world sheet given in terms of a group valued map g : Σ → G by e ± = g −1 ∂ ± g = e a ± T a and T a , with a = 1, · · · , D and D = dimG, generate the algebra g. For later use we denote the vector fields dual to these one-forms as v a = v a i ∂ i .
The matrix E 0 = g 0 + b 0 contains D 2 constant entries (which one could promote to fields depending on external spectator coordinates) expressed in terms of a symmetric and a skew-sym metric matrix, g 0 and b 0 , respectively. π is a Poisson-Lie structure compatible with the group multiplication of G [36], i.e. 2 π g 1 g 2 = π g 2 + Ad g −1 thus turning the pair (G, π) into a Poisson-Lie group. The Poisson structure induces a Lie algebra structure on the dual vector space g * , which we will denote byg, and we indicate the corresponding structure constants byf ab c . The two algebras can be put together so as to form a (classical) Drinfel'd double d = g +g, a 2D-dimensional Lie algebra equipped with an ad-invariant inner product ⟪•, •⟫ for which g andg are maximally isotropic subalgebras [4]. Whilst the action (2.1) is not invariant under global G transformations acting from the left, the properties of π ensure that the corresponding would-be-Noether currents j, whilst not conserved, still obey a modified non-commutative conservation law [4] d j a =f bc a j b ∧ j c . (2. 3) It is this peculiar property that is called a Poisson-Lie symmetry. By introducing local coordinates x i on G, one can extract from the action (2.1) a metric g ij and a two-form b ij . For later use we also define local coordinates on the double (x i ,xĩ) on respec tively G and G in D = exp d and the corresponding doubled partial derivatives ∂ I = (∂ĩ, ∂ i ). In general, π will contain rather involved functional dependence on these local coordinates and the inverted matrix that defines (2.1) will be complicated, thus rather obscuring the elegance of the action that is indicated by the equation of motion (2.3). Instead, a clearer view is obtained by working with the generalised metric which combines the metric and B-field in a unified object. We will come back to this point in the next subsection.
An example of a Poisson-Lie symmetric model that will play a particular role in this letter is the well-established two-parameter integrable model known as the bi-Yang-Baxter theory [11,37], which is associated to the Drinfel'd double d = g C , and for which Here η and ζ are real valued deformation parameters, t is the tension for the σ-model, g ∈ G is a group element, κ is the Killing form, R ab = −R ba = κ ac R c b solves the modified classical Yang-Baxter equation and R g = Ad g −1 · R · Ad g . The essential properties enjoyed by R are that it obeys R 3 = −R, it acts by swapping positive and negative roots, and projects to zero directions in the Cartan subalgebra of g. In fact, the R-matrix completely captures the Lie algebra structure of the dual bi-algebrag to g inside d through the identityf ab c = 2R d[a f dc b] .
Suppose now g admits a subalgebra h such that g = h + m is a symmetric space decompo sition. That is, we assume the presence of an involutive Lie algebra automorphism σ grading g, whereby we identify the ±1 eigenspaces g (0) and g (1) with h and m, respectively. We thus split the generators accordingly T a = T a , T α such that m = span(T a ) and h = span(T α ). A natural suggestion for a Poisson-Lie σ-model on the coset G/H (which we will always take to be reductive) is to restrict the indices in (2.1) to run over m, and so we consider However, a priori, the degrees of freedom entering this action still contain those corresponding to the subgroup H and thus, without imposing further constraints, (2.5) does not provide a description of the coset G/H. This is remedied by demanding that the action develops a local gauge symmetry under the action of H from the right which serves to eliminate the unwanted degrees of freedom. A short calculation shows that under an infinitesimal transformation this is the case provided that [38] 0 =f ab γ + E −1 ad The doubly deformed model we will introduce will demand us to specialise to the situation wherẽ f ab γ = 0. This can be stated in a more invariant fashion as demanding that h is coistropically 3 embedded in g. A useful consequence of this condition is that the coset-coset components of the bi-vector evaluated on a subgroup element vanish, i.e. π ab h = 0 for h ∈ H. As a consequence, under the parametrisation g = mh with h ∈ H, we have from the multiplication law (2.2) that This makes it immediately clear that, provided E 0 is H-invariant, the action (2.5) only depends on the coset representative m in the decomposition g = mh.

E-model for group manifolds
In this section we will first define E-models for group manifolds and introduce the central concept of generalised frame fields. The latter enables one to access the σ-model geometry directly from the corresponding E-model.
The E-model [6][7][8] is a theory of currents J = T A J A (σ) = g −1 ∂ σ g valued in the loop algebra of d which originate from the embedding map g : Σ → D of the string world sheet into the Drinfel'd double and T A are the dim D generators of d. The dynamics is generated by the in which the eponymous operator E : d → d is an involution, E 2 = 1, that is self-adjoint with respect to the inner product ⟪·, E·⟫ = ⟪E·, ·⟫. This involution can be specified by D 2 parameters which are associated to those of the Poisson-Lie σ-model by where g 0 and b 0 are, respectively, a constant symmetric and antisymmetric matrix and H AB is the associated generalised metric. Through their equal-time Poisson brackets these currents realise a classical current algebra in which F AB C are the structure constants on d. Accordingly, we find the equations of motion Taking into account that E is an involution, we could also decompose the currents into chiral and anti-chiral parts and rewrite (2.11) as Since EJ ± = ±J ± , only half of the components of J + and J − are independent. Therefore, although the E-model apparently depends on 2D degrees of freedom contained in g, the first order equations of motion allow half of them (those associated to the subalgebrag) to be eliminated on-shell to yield second order equations for the rest (those associated to g). In this way the dynamics of the theory specified by the σ-model (2.1) on the target space M = D/ G is recovered.
The Hamiltonian for a non-linear σ-model can be expressed in terms of a generalised metric H IJ as (2.14) in which the canonical momentum is given by p i = (g ij ∂ τ x j − b ij ∂ σ x j )/2πt and where (g ij , b ij ) is the geometric data entering the σ-model. Here the indices I denote that the fields act on the gen eralised tangent space of M and one has the usual canonical Poisson brackets {x i (σ), p j (σ )} = δ i j δ(σ − σ ). Given the structure of Ham H , it is natural to introduce some generalised currents J I = ∂ σ x i , 2πp i taking values in the generalised tangent space of the target space M. By virtue of the canonical Poisson brackets for (x, p), they can be shown to obey where η is again the O(d, d)-invariant metric. The same choice of letter for these currents is not accidental as they can be related to the E-model currents J A by introducing generalised frame fields E A I such that Here, and henceforth, η AB and η IJ are used to raise and lower "doubled" algebra and target space indices respectively. The generalised frame field E A I is the generalised geometry analog of the left invariant Maurer-Cartan forms e a i on the target space group manifold M = G. For each value of the algebra index A, E A I defines a generalised vector -a section of the generalised tangent bundle E = T M ⊕ T * M -comprising of a vector field and a one-form (whose components in a coordinate basis are E A i and E Ai , respectively). For a Poisson-Lie σ-model on a group manifold the generalised frame fields are [39] and This relation is well known in the context of generalised parallellisable spaces and its connection to the E-model was recently studied [40]. Armed with such a generalised frame field, the elegant results of Alekseev and Strobl [41] show that one can indeed, starting from the canonical Poisson-brackets for p i and x i appearing in (2.14), derive the Poisson brackets for the currents J A as Upon substitution of (2.19) into (2.20) one does consistently re-obtain the Poisson-brackets (2.10).
In the remainder of this section, we will take the opportunity to work out and clarify, following [14,42], the relation between the σ-model and the E-model description. Although we will not aim to encompass all possible E-models, the construction below will cover most of the relevant models studied to date.
The equations of motion for the σ-model on G are well-known to be expressed in terms of some currents j ± ∈ g; the latter, however, are not in general trivially embedded in their doubled (anti)chiral counterparts J ± . To keep track of this, we introduce a surjective map S : g × g → d, called the S-map, satisfying the following axioms for all x, y ∈ g S(x, y) = S(x, 0) + S(0, y) , Given the splitting (2.12), we are allowed to identify Particularly interesting are E-models where the time evolution (2.13) can be recast as the flatness Here we are using the generalised Lie derivative acting on sections of the generalised tangent space. For a pair of generalised vectors U = u i ∂i + µidx i and V = v i ∂i + νidx i the derivative takes the form where L is the conventional Lie derivative.
of a g C -valued Lax connection L(z) = L + (z)dξ + + L − (z)dξ − with spectral parameter z. In fact, (2.25) guarantees their classical integrability. For example, one can consider a class of models whose equation of motion and Bianchi identity have the form where ν ± ∈ C and O ± is an endomorphism of g. Many different models like, for instance, the principal chiral model, the Yang-Baxter model (possibly with a Wess-Zumino term) and the bi-Yang-Baxter model fit into this scenario, upon choosing the appropriate form for ν ± and O ± . They all admit the Lax representation with For each of these, in order to recover this result from the E-model, a further condition replicating A similar construction has already been considered, albeit in a slightly more abstract fashion, in some explicit cases, for instance for the bi-Yang-Baxter model in [43] where d = g C .

E-model for coset spaces
On coset spaces the situation is similar albeit the derivation of the Poisson brackets is more involved. An extensive and general analysis of this setting was recently carried out in [1]; here, we aim at specialising the construction therein to the case at hand. We take the generalised coset space 5 M = H\D/ G and label its generators as T A = ( T a , T a ), with M = span(T A ), where this partition of generators respectively spans m * and its dual m. In this case we wish to restrict the Hamiltonian to depend only on the currents J = T A J A (σ). To better fit into this formalism, the condition (2.6) can be equivalently rewritten as 29) and the Jacobi identity of d ensures that (T α ) A B = F αA B generate h. Hence, we see that the gen eralised metric is ad-invariant under H-action. As a result the model develops a gauge symmetry and the currents along the subgroup directions become non-dynamical, invoking constraints. Just as one can obtain conventional frame fields for a normal coset G/H by reducing and gauge fixing the frames defined on the group G, we can obtain generalised frame fields on the generalised coset by adopting a similar process. A particularly simple example is provided by the case of H being a coisotropic subgroup of a Poisson-Lie group G (recall from the discussion around eq. (2.6) that this requiresf ab γ = 0) since then the Poisson structure on G descends directly to one on G/H [36]. This is sometimes called a Bruhat-Poisson structure and denoted by π B . Paralleling the discussion for E-models on group manifolds, the σ-model geometry for coset spaces can be obtained by introducing the generalised frame fields given by [1] Here e a are the (gauge-fixed) coset components of the left invariant one-forms and v a are their Poincaré dual vector fields. They arise naturally from the decomposition of the left-invariant Maurer-Cartan form m −1 dm = T a e a + T α e α where m ∈ M parameterises the generalised coset space. Similarly to conventional coset spaces, we need to invoke an h-valued generalised con nection Ω α (introduced in [1]) entering the generalised frame field on the coset mediating the H-compensating transformations and taking care of the gauge invariance, In order to find the frame algebra obeyed by the generalised frame field, we make use of the machinery in [1] to obtain where Here the currents J A are defined in the same spirit as in (2.16), where we now restrict indices to run over generalised coset generators, A → A, and coordinates, I → I. In contrast to the case of a group manifold, the F ABC appearing in (2.33) are no longer constants but instead given by where Enforcing the corresponding Jacobiator one obtains the consistency requirement where we have introduced the derivative operator D A = E A I ∂ I obtained by contracting the doubled partial derivative with respect to the generalised coset coordinates with the frame fields restricted as above. This condition is automatically fulfilled using the Ω-compensator together with the frame fields (2.30). To completely characterise the system we shall also require the Poisson brackets for the compensator The Jacobiators involving the combinations (J , J , Ω), (J , Ω, Ω) and (Ω, Ω, Ω) vanish identi cally as well.
The equations of motion can be cast in a form similar to (2.13) by introducing the quantity Note that the two objects J ± = J A ± T A are defined as in eq. (2.12). While the equations of motion for B ± read where P h is a projector from d to the isotropic subgroup h.
Let us now construct the S-map explored above for coset E-models when H is a coisotropic subgroup and G/H a symmetric space. This latter assumption is sufficient to prove that the dual algebrag equipped with the Lie bracket induced by the R-matrix is a symmetric space as well. As a consequence, the structure coefficients F ABC vanish and with them so does the projection along M of the bracket [J − , J + ]. In such a scenario we wish to express the currents J ± valued in M in terms of the m-valued currents j ± using the S-map as before. To this end, suppose that S : g × g → d satisfies the two constraints (2.21) and (2.22). Assume now it admits a restriction to the coset m such that ImS(m, m) = M. The constraint (2.23) is preserved, provided that we now interpret, using the coisotropy property of h, E as the restriction of the generalised metric to the generalised coset, i.e. E : M → M, and take x, y ∈ m. In addition, we impose that for all The field equation (2.37) is then equivalent to while the Bianchi identity reads Such theories are classically integrable and their dynamics is encoded in the flatness of the Lax connection To give a flavour of S in a case relevant to the discussion ahead, the map for a Yang-Baxter deformation of CP n turns out to be 6 , for every x, y ∈ m, where η is the deformation parameter and t the tension of the σ-model. 6 We assume that R here is a restriction of the Yang-Baxter matrix to the sole coset directions, i.e. R : m → m.
It acts on an element Analogously, κ is the restriction of the Killing form to the generators of m.

Integrable deformations of CP n
In this section we shall put the general machinery presented thus far to use. We start by consid ering a double deformation already introduced in [25] consisting of the deformation parameters η and ζ. However, as it transpires from the study of its integrability structure, the second pa rameter ζ does not play a role, for it can be reabsorbed in new effective tension and deformation parameter. At the geometrical level we can equivalently construct a diffeomorphism undoing the deformation introduced by ζ. Remarkably though, the deformed geometry of CP n spaces can be elegantly described in terms of generalised Kähler geometry. All the geometrical aspects will be the subject of Section 4, while we devote the present one to the study of its integrability.

σ-model description
Let us first return to the constraint of H-invariance required on E −1 0 | m in eq. (2.6) (or equivalently eq. (2.29) on the generalised metric). We wish to consider specific choices of (d, g, h) for which the admissible H AB allows for multiple free parameters. A model with such features was already identified in [25]. There it was shown that H-invariance of its generalised metric implied the constraintf (3.1) When this condition holds, h is said to be a coisotropically embedded subalgebra, as introduced above in Section 2.1, and the Poisson structure on G descends to one on G/H making it a Poisson homogenous space 7 .
In what follows it will be crucial to refine, motivated by the goal of integrability, the set of coset spaces to Hermitian symmetric spaces, implying that R a c R c b = −δ a b i.e. the R-matrix descends to a complex structure (see Appendix A for a more precise treatment). The canonical examples of such a set-up are the complex Grassmannians for which G = SU (n + m) and H = S(U (n) × U (m)), of which CP n is a particularly interesting subcase 8 . Thus, assuming that G/H is an Hermitian symmetric space with H a coisotropic subgroup, the σ-model reads in which R m = Ad m −1 · R · Ad m and where, to match notation elsewhere, we introduce projectors P i onto g (i) in the symmetric space decomposition and let κ ab = T a , T b . Because of gauge invariance, the action (3.2) depends only on the coset representative m in the parametrisation g = mh with h ∈ H. 7 Notice that this condition fails to hold (see e.g. [44] or Appendix A) for g = so(n + 1) and h = so(n) for n > 2 and so we can't reduce to a theory on S n in this way (for n = 1 we have S 2 CP 1 , which is amendable of such deformation, as we shall see). 8 Regarding CP n σ-models we are making statements only about classical integrability; one should anticipate that quantum integrability necessitates the inclusion of appropriate fermionic content [45,46]. Fortunately our results point to how this should be done since they will be compatible with N = 2 world sheet supersymmetry.

Weak and strong integrability
Proving that a model is (classically) integrable in the weak sense amounts to find a Lax connection whose flatness implies the equations of motion and Bianchi identity. To this end we introduce the currents 9 where k is the combination of parameters solving k 4 − (1 + η 2 − ζ 2 )k 2 − ζ 2 = 0. The Lax connection can then be shown to be As for strong integrability, a sufficient condition is to ascertain that the Lax matrix L(z) = L + (z) − L − (z) obeys the Maillet algebra [35,48] for a specific choice of the matrices r and s. These two are, respectively, the skew-symmetric and symmetric part of a solution R of the mCYBE for the loop algebra of g. Of particular interest are the models for which R is fully specified by a model independent R 0 , usually expressed in terms of the graded projections of the Casimir operator 10 C (ij) of g, multiplied by a scalar function of the spectral parameter ϕ(z), known as the twist function [49]. In this case and when considering coset spaces, r and s assume the form (3.6) (3.7) The E-model formalism facilitates the analysis; we first introduce the Lie algebra valued quanti ties where we take g = mh and p the canonical momentum. The advantage of these definitions is that the Lax matrix can be written as a function of X and Y . In turn, these are conveniently embedded into a generalised vector via Z A = 2π(tY a , X a ). The crucial observation is that the latter is related to the currents J A introduced in Section 2.2 by a constant O(d, d) transformation β As a result, the Poisson brackets for Z , and thus the ones for L(z), can be inferred from the ones for J already given in (2.10). Eventually, the Poisson brackets for the Lax matrix can be used to check if the Maillet algebra (3.5) is satisfied. As it turns out, our model fits into the class of models with twist function, where we find (3.10) Upon identifying new deformation parameter and tension according tõ one can see that the twist function above coincides with the one for the single Yang-Baxter deformation already present in the literature [47]. It is tempting to infer, judging from this analysis, that the second parameter ζ is not affecting the model. Nevertheless, given the exis tence of transformations not affecting the twist function [42] and yet yielding to a non trivially deformed target space, a more detailed geometric check is needed. We will explicitly construct the diffeomorphism removing ζ from the deformed metric in the next section.

Renormalisation group flow
A useful cross-check of the above result comes from analysing the renormalisation group flow for the effective parametersη andt. Using the doubled formalism introduced in Section 2.3, the renormalisation group flow can be written as where R AB denotes a projection of the generalised Ricci tensor for a generalised symmetric space whose form can be found in eq. (3.36) in [1]. An equivalent form of the flow equation, which is the one we shall eventually make use of, is given by Of the three parameters η, ζ and t entering eq. (3.2) there are two RG invariants which can be chosen to be ηt and 1+ζ 2 +η 2 ηζ . The remaining non-trivial flow equation is obtained as 11 dη d log µ ∝ ηt(1 − ζ 2 + η 2 ) . (3.14) One can prove thatη andt, as defined in eq. (3.11), obey the same flow eq. (3.14) (upon setting ζ = 0). In particular, the fact that ηt =ηt implies that the latter is again an invariant.

Generalised Kähler structure of deformed CP n
In this section we show how the backgrounds presented in the previous section are, notwithstand ing their apparent complexity, exemplars of generalised Kähler manifolds.

Target space geometry
We now turn to the curved target space geometry. One can extract a conventional metric g and NS two-form field b either from directly evaluating the σ-model action of eq. (3.2) or equivalently by dressing the generalised metric with the generalised frame fields which we recall are given by 12 in which we have introduced a coset representative m with m −1 dm = e a T a + e α T α . Here π B is the Bruhat-Poisson structure on the Poisson homogenous space G/H, which we can express as where we recall that R m = Ad m −1 · R · Ad m . By assumption the target space is in addition equipped with a complex structure, an Hermitian metric (e.g. the Fubini-Study metric in the case of CP n ), and a Kähler two-form that we denote respectively by It will be convenient to observe that in this set up we can introduce a more general bi-vector consisting of the linear combination of the Bruhat-Poisson structure and the inverse of the Kähler-Fubini structure, forming a so-called Poisson pencil, which, for CP n at least 13 , obeys the Schouten identity and defines a genuine Poisson structure. After some manipulations one finds that the deformed geometry encapsulated by g and b can be expressed in terms of G and π τ as in which the Poisson pencil parameter is fixed to τ = 1 + ζ η . Let us emphasise that, despite the elegant form of (4.6), in terms of explicit coordinate expressions these become rather intractable. 12 With respect to eq. (2.30) we have introduced the factor ηt needed to ensure the equivalence between the Poisson-Lie model (2.5) and the action (3.2). 13 In general πτ does not obey the Schouten identity. However, as proven in [51], it does on any Hermitian symmetric space (see Appendix A for a thorough discussion).

From double to single parameter deformation
Let us start by recalling the Fubini-Study metric on CP n in which the coordinates z i are holomorphic with respect to a complex structure J = idz i ⊗ ∂ z i − idz i ⊗ ∂z i and |z| 2 = z izi . More precisely, we put ourselves in the patch, often called the largest Bruhat cell, where the first homogeneous coordinate is not vanishing. Let us now introduce coordinates (x i , φ i ) better adapted to the deformed geometry; they are defined by where X = i x i . With respect to these, the Poisson-Bruhat structure reduces to As the strong integrability and RG-flow analysis suggested, the second parameter ζ appears to be redundant. Indeed, we have explicitly checked up to n = 6 that its effect induced on metric and 3-form H = db can be removed using new coordinatesx i defined as where α is given by α 2 = k 2 − 1 −η 2 and leaving the angles φ i untouched. Although we do not dispose of a general proof, we believe we can safely conjecture its existence for all n. Given this diffeomorphism, we will henceforth drop any dependence on ζ setting ζ = 0. Nevertheless, the geometry is still best described using the Poisson pencil π τ as in (4.5) by fixing τ = 1.
Even with a single parameter and in adapted coordinates, the forms of the deformed metric and B-field result in quite involved expressions and we will thus refrain from showing them here explicitly. Rather, we will argue in the next section that the resulting geometry is bi-Hermitian and we will provide canonical formulas for the geometric objects of interest.

Generalised Kähler structure
We will now study the generalised Kähler structure of the above integrable models. We first present a rapid recap of the salient details of this geometry [52,53], before examining the struc ture on deformation of CP n keeping n unspecified. To progress further we give some explicit expressions for the cases of n = 1, 2, 3 since they have the most appeal for potential embeddings within a string or type II supergravity set-up.

A précis of Generalised Kähler Geometry
Demanding that a non-linear σ-model admits an N = (2, 2) extended supersymmetry requires that the target space be bi-Hermitian [34,54,55]. That is, the metric should be Hermitian with respect to two complex structures J ± each of which is covariantly constant with respect to the torsionful connections 0 = ∇ (±) J ± = (∂ + Γ ± H) J ± with H = db. As shown by Gualtieri [52], bi-Hermitian geometry is equivalent to generalised Kähler geometry in which the generalised metric can be decomposed in terms of two commuting generalised complex structures as E = J 1 J 2 . The map between these two notions is with ω ± = J ± g. This extends the notion of Kähler geometry and when true the entire geomet rical data (g, b, J ± ) can in principle be encoded in a single function K of superfields called the generalised Kähler potential. However, explicit forms for such K are generally rather challenging to extract. To progress in this direction it is useful to first note that one can construct three Poisson structures 14 [56] The types of supersymmetric multiplets required to furnish an N = (2, 2) action can be extracted from these. Namely, chiral superfields parameterise ker(J + − J − ) = ker π − (4.12) whereas twisted chirals are needed to parameterise The remaining directions i.e. (ker([J + , J − ])) ⊥ corresponding to the symplectic leaves of σ are to be parametrised by semi-chiral superfields [56]. A useful alternative description is based on the construction of pure spinors Ψ i whose an nihilator space under the Clifford action coincides with the (+i)-eigenspace of J i . Using the Clifford isomorphism we can view Ψ i equivalently as either chiral/anti-chiral Spin(D, D) MW spinors or as the formal sum of either even or odd degree differential forms and we go back and forth freely between these presentations.
A key challenge in establishing the generalised Kähler potential is to find appropriate coordi nates. We are first required to identify sets of coordinates (Q, P ) or (q, p) that are holomorphic diagonalizing J − or J + respectively, as well as their conjugates (Q,P ) and (q,p). It is a trivial matter to check that J t ± σJ ± = −σ, i.e. that σ splits into σ = σ (2,0) + σ (0,2) with respect to either complex structures. Invertibility, however, is not necessarily guaranteed: σ is only invertible on the leaves of its symplectic foliation. In what follows, when we write σ −1 , one should understand it to be the inverse of this restriction of σ. Because σ is a Poisson structure dσ −1 = 0 has to hold. In general σ −1 will inherit from σ the decomposition σ −1 = σ −1 (2,0) + σ −1 (0,2) and the holomorphic coordinates should be such that it is brought to the canonical form σ −1 = dq m ∧ dp m + c.c. = dQ m ∧ dP m + c.c. . (4.14) The crucial point is that the transformation between (p,p, Q,Q) and (P,P , q,q) is canonical with a (real) generating function K(P,P , q,q) such that It is this generating functional that becomes identified with the generalised Kähler potential. Thus extracting K can in general be hard and cumbersome: first one has to obtain p m and Q m and then integrate the above equations to determine K.

CP n
We will start by studying the invariance property of CP n as a coset manifold, taking G = SU (n + 1) and H = S(U (n) × U (1)). In this case we obtain the branching rule of G irreps to H. Eventually our goal is to find forms which are invariant, i.e. singlets, under the holonomy group H. For SU (n) we know that trivial representations only arise in the tensor products n×n and n×n. Thus, with the decomposition (4.16), we find that there are two SU (n) invariant two forms: a symmetric one which is the restriction κ of the Killing form on G to the coset and an antisymmetric one which is ω, the Kähler form. Both are invariant under SU (n) and U (1). Furthermore there are two SU (n) invariant n forms Ω and Ω with U (1)-charges n and −n, respectively. There also has to be an invariant R ab but the only invariant two-form we found is ω. Thus, we conclude that R ab = ω ab , in perfect agreement with the interpretation of R a b as a complex structure on the algebra.
We can now observe that the generalised metric with flat indices given in eq. (4.1) (now with ζ set to zero) admits the decomposition E = J 1 J 2 with such that Thus upon dressing these flat space quantities with the generalised frame fields (4.2) we see that the target space geometry is indeed generalised Kähler with It is now easy to show that, if we introduce the quantities Q ± = 1 ± bg −1 = 1 ∓ ηGπ τ , the objects appearing in the map (4.11) are given by (4.20) The three Poisson structures π ± and σ are extracted as Let us briefly study the superfields associated to these structures. For π + , its kernel is isomorphic to the kernel of 1+(ηπ τ ω) 2 which, since (ηπ τ ω) 2 is positive definite, is trivial. Hence, no twisted chiral multiplets are present. The kernel for π − is better studied in Fubini-Study coordinates of the largest Bruhat cell. Here the complex structure is diagonal and the expression (4.21) for π − amounts to selecting the diagonal blocks of π B which, in this patch, turns out the complex conjugates of one another. Each one of these blocks is a n × n dimensional matrix and, therefore, has vanishing determinant for odd n. In particular, its null space is parametrised by a single vector. In the even case, it turns out that the determinant is non-vanishing, implying a trivial kernel. In summary, when n is odd we have two vectors generating the kernel of π − and, thus, a single chiral superfield. We therefore end up with (n − 1)/2 semi-chiral multiplets plus a single chiral multiplet in the odd case and n/2 semi-chiral multiplets in the even case.
The same results can be obtained employing pure spinors; we recall that the type k 1,2 of a pure spinor Ψ 1,2 is related to the kernel of the Poisson structures via the relation dim ker π ∓ = 2k 1,2 . (4.22) To compute these pure spinors it is efficient to first compute the corresponding flat space equiv alents Ψ i corresponding to the complex structure in eq. (4.17) and, using the spinorial repre sentation of the generalised frame fields S E , later build Ψ i = S E Ψ i , where S E = S π S e is the corresponding spinorial representatives of the decomposition in (4.2).
We can now check how the exterior derivative acts on these two spinors. Pulling the exterior derivate through the spinor action of the generalised frame field results in the F -twisted derivative with the generalised dilaton d = Φ − 1/4 log det g. Note that because we are considering a symmetric space there is no F ABC contribution. Using the definition (2.31) of the generalised connection we eventually obtain where the index 0 refers to the component of e α and the bi-vector along the holomorphic U (1) in the branching (4.16). This proves that we have a generalised Kähler manifold and at the same time that the three Poisson structures we presented above are indeed Poisson structures. Finding the generalised Kähler potential K for a deformation of CP n is complicated, at least for generic n. We thus defer to the next section the explicit computation in the n = 1, 2 cases, whilst we devote the present paragraph to some more general reasoning. To find the (p, q) and (P, Q) coordinates explicitly we exploit the fact that there are n Killing vectors which leave σ −1 invariant (considering for simplicity here the case relevant to CP even for which σ is invertible). The coordinates (x m , φ m ) introduced in eq. (4.8) are adapted to this such that the Killing vectors are simply given by the ∂ φm . We can select the holomorphic (with respect to J ± ) part of σ −1 by acting with a projector such that dq ∧ dp + dQ ∧ dP = σ −1 we obtain the momentum maps which, together with the one-forms dφ m dual to the isometries, form a basis of one-forms. A symplectic form σ −1 which satisfies (4.28) has to have the form where a mn = ι ∂ φ m dµ n with da mn = 0 . Furthermore, σ −1 has to be closed. This implies that the only free parameter b mn has to be constant like a mn . To fix b mn , we just have to match the left and right hand side. As result, we find that where dq m and dp m are linear combinations (with constrained coefficients) of dφ m and dµ + m . The same holds for dQ m and dP m but with respect now to the linear combination built from dφ m and dµ − m . So the procedure is simple in principle: first integrate the moment map to find the µ m and take appropriate linear combinations µ and φ to define the canonical coordinates. Then find the generating function K by integrating the canonical transformation of eq. (4.15).

CP 1
It is a well-known fact that every two-dimensional complex manifold is Kähler; as such, the deformed CP 1 geometry is completely determined by the standard (i.e. non generalised) Kähler potential. In fact, one can further notice that, given the dimensionality, the B-field is always pure gauge and thus negligible. As for the patch, we put ourselves in the largest Bruhat cell where the homogeneous coordinate Z 0 = 0 and introduce the holomorphic coordinate z ≡ Z 1 /Z 0 . The Kähler potential is where we notice that the η → 0 limit is non-singular and yields K FS , i.e. the undeformed Fubini-Study Kähler potential for CP 1 , K FS = log(1 + |z| 2 ).
As CP 1 is Kähler, J + = J − , and π − vanishes. In turn there is a single set of complex coordinates diagonalising J ± expressed by and its conjugate, where (x, φ) were introduced in (4.8), and 15 The pure spinors are easily found to be, up to an overall normalisation, where the two-form ρ is best expressed in new coordinates on CP n defined by such that Ψ 1 starts with a one-form and is thus said to be of type 1; on the other hand, Ψ 2 starts with a zero-form and is thus of type 0. In the language of N = (2, 2) supersymmetry, we analogously say that the geometry is described by a chiral superfield.

CP 2
CP 2 is the first case where we can study a non-trivial generalised Kähler potential and give a rather nice explicit presentation thereof. In the largest Bruhat cell, the family of Poisson structures is given by where the general expression for π B was introduced in (4.9) and the Kähler form is constant in these coordinates. For the η-deformed geometry we simply need to set the parameter appearing in the Poisson pencil in eq. (4.5) to τ = 1 in the above. A basis for the holomorphic generalised vectors of J 1 is given simply in the FS coordinates as {−itηz 1z2 ∂z 1 + dz 2 , itηz 1z2 ∂z 2 + dz 1 , ∂ z 1 , ∂ z 2 } , (4.39) from which it is easy to see that the corresponding pure spinor Ψ 1 can be given as The holomorphic generalised vectors of J 2 are best expressed in the (x, φ) coordinates for which π τ is simple and are given by The spinor corresponding to J 2 is again expressed by where ρ is defined as in (4.37) with n = 2. Both of these pure spinors start with a zero-form contribution and hence are said to be of type 0. In terms of supersymmetry this implies that the deformed geometry is characterised by a single set of semi-chiral superfields.
Notice that here, as in the CP 1 case, both the pure spinors are closed; however, we have made a choice in fixing the normalisation such that the zero-form component is a constant equal to one. We can also consider the normalisation under the Mukai pairing || Ψ i ||= Ψ i ∧ σ( Ψ i )| top in which the σ operation in this case simply switches the sign of the two-form component. We have that Should we scale the pure spinors such that they have equal normalisation, then they would no longer be closed. The geometry is hence not generalised Calabi-Yau 16 . Now, a first step in computing the generalised Kähler potential is to find the holomorphic coordinates of J ± , that is, to identify p(z,z) etc. such that σ −1 = dp ∧ dq + dp ∧ dq = dP ∧ dQ + dP ∧ dQ . Using the sympletic moment map associated to U (1) actions as described in the previous section one finds for p, q (withp,q given by standard complex conjugation) where the angles χ 1,2 are a generalisation of the one previously introduced and µ and β follow the definition in (4.34). In particular, µ is a coefficient needed to ensure that σ −1 has the correct form (4.45). There is a simple relation between these p, q and P, Q, namely Letting the generating function K(P,P , q,q) = −(P q +Pq) + K 1 (P,P , q,q) , (4. 49) we require, in accordance with (4.15) and (4.48), that A closed form for the potential in terms of the angles χ i can be given in terms of the parametric integral I α (y) = y cot y + α 2 dy = 2 y log 1 − e i(α+y) − iLi 2 e i(y+α) − iy 2 2 , (4.51) such that To complete the specification of the potential one needs to express the χ i in terms of (P,P , q,q) which can be done implicitly via the relations e |p|/|µ| = sec(β) csc(χ 2 ) sin(β + χ 1 − χ 2 ) , e |q|/|µ| = csc(β − χ 1 ) sin(β + χ 1 − χ 2 ) . (4.53)

CP 3
We could proceed in a similar direction with the case of CP 3 though explicit expressions become unwieldy rather quickly. Let us instead emphasise an important difference compared to the case of CP 2 , which can be seen readily from the pure spinors. Corresponding to the generalised complex structure J 1 we have (in FS coordinates) which is now (generically) of type 1 implying that there is a chiral multiplet. Indeed using eq. (4.12) one establishes that the vectors in ker(J + − J − ) are dual to the one-form component of Ψ 1 . The pure spinor corresponding to the generalised complex structure J 2 is given exactly as in the other cases i.e. Ψ 2 = exp ρ , with ρ defined as in eq. (4.37) with n = 3. With this normalisation we see that d Ψ 1,2 = 0. Since Ψ 2 is of type 0 we conclude that the geometry of the deformed model on CP 3 requires a chiral multiplet and a semi-chiral multiplet.

Conclusions and outlook
In this work, using the tools of Poisson-Lie non-linear σ-models on generalised coset spaces G/H [1], we have described a specific but particularly striking class of examples in which G/H were Poisson Hermitian spaces. Upon constructing an integrable Yang-Baxter deformation of these, we showed that their target space is described by generalised Kähler geometry. We discussed in detail CP n as a prototypical example and, for the case of CP 2 , gave an explicit formulation of the corresponding generalised Kähler potential. We filled a gap in the literature by showing that a previously conjectured two parameter deformation for CP n is indeed integrable but coinciding with the already well-known Yang-Baxter deformation of coset spaces. A background motivation for this work was to investigate integrable deformations of AdS 4 × CP 3 with the aspiration of identifying quantum group deformations in the ABJM model. At first sight the corresponding geometry is rather unattractive but in this work we have elucidated many of the key features. A complete analysis would of course require furnishing the geometry with appropriate RR fields and investigating the fermionic sector. Whilst one might "boot-strap" an RR sector, an approach done for the case of Poisson-Lie models on groups in [40], ultimately it would be desirable to extend the considerations to super-cosets [19][20][21][22][23].
In this letter we only considered coset spaces for which the gauge group is coisotropic, as these naturally solve the invariance constraint, leaving the construction of other holographically relevant coset spaces open. Moreover, the explicit examples taken into account here were based on quotients in which the Drinfel'd double was d = g C . The incorporation of λ models requires the more general case of d = g + g; our general tool kit [1] accommodates this scenario and so it would be interesting to explore if there can be some underlying generalised Kähler structures in the λ-deformations of G/H-WZW models.
A fruitful further line of investigation would be to consider the presence of D-branes in these geometries and understand how the elegant characterisation in generalised Kähler geometry can relate to integrability preserving boundary conditions as in e.g. [57]. Beyond such direct follow up tasks, this work suggests a number of interesting broader questions, and we list a few here with the hope of returning to them later: • How generic are the constructions of Poisson-Lie models on groups and coset spaces; how much of the landscape of integrable σ-models do they capture?
• We saw some interplay between integrability and extended supersymmetry. However one need not expect all N = (2, 2) models to be integrable, but which are? How is this reflected at the level of superspace?
A subalgebra h of a Lie bialgebra g is called coisotropic if its annihilator h ⊥ , i.e. the space of functionals ξ ∈ g * such that 17 ξ, x = 0 ∀x ∈ h, is a Lie subalgebra in g * [58]. Whenever G is compact and semi-simple it is naturally endowed with a non-degenerate pairing κ, the Killing form, constituting a canonical isomorphism between g and its dual g * ; this fact, together with the orthogonality of m and h with respect to κ, can be used to show that m h ⊥ * . Finally, this duality implies that h ⊥ being a subalgebra is in fact equivalent to [m, m] R ⊂ m or, equally, to [m, m] R | h = 0, which is just (A.1).
Verifying that H is coisotropic, or better said coisotropically embedded, in G is no trivial task. One possibility is to exploit the method introduced in [44]; however, we shall pursue a different approach which better reflects the (Poisson-Lie) group theoretic origin of our system.
A subgroup H of a Poisson-Lie group G is called a Poisson-Lie subgroup if the annihilators h ⊥ are an ideal in g * [58]. Since any ideal is a subalgebra, any Poisson-Lie subgroup is automatically a coisotropic subgroup (but the converse is in general false). Moreover it will be useful for us to notice that, whenever H is a Poisson-Lie subgroup, the reduction of the Poisson structure π defined on G to its restricted counterpart on G/H is unique and gives rise to the so-called Poisson-Bruhat structure π B [36].
Another relevant fact is that every coadjoint orbit O of a Poisson-Lie group arises from the quotient of the latter with a Poisson-Lie subgroup [59]. Thus coadjoint orbits of (compact semi-simple) Poisson-Lie groups can serve useful examples of target spaces in the theories we are considering.
Such coadjoint orbits can be equipped with a closed, non-degenerate, symplectic 2-form ω usually known as the Kostant-Kirillov-Souriau (KKS) form, where non-degeneracy implies that each orbit is an even-dimensional symplectic manifold. It can be further shown [60,61] that this symplectic structure is actually a Kähler form and thus every O is a Kähler homogeneous space. Now, in the light of the above, on each orbit we evidently have at our disposal two different Poisson structures, namely π B and ω −1 , since dω = 0 implies [ω −1 , ω −1 ] s = 0 (where [·, ·] s is the Schouten bracket). Given two Poisson structures π 1 and π 2 one could try and combine them so as to create a third one, π(τ ) = π 1 + τ π 2 , where τ ∈ R. In general, the Schouten bracket [π(τ ), π(τ )] s does not vanish but whenever [π 1 , π 2 ] s = 0 (and in this case π 1 and π 2 are said to be compatible) it obviously does. If such a condition is met then π(τ ) is referred to as a Poisson pencil. On the coadjoint orbit O, checking whether [π B , ω −1 ] s = 0 is not straightforward and might require an explicit calculation. However, as shown in [51], on any (compact) Hermitian symmetric space the two Poisson structures π B and ω −1 are compatible and hence give rise to a pencil.

Poisson pencils on Hermitian symmetric spaces
Thus we are particularly interested in finding coadjoint orbits which are also Hermitian sym metric spaces so that they will be naturally endowed with a Poisson pencil. The first step is to observe, following Kirillov [62], that any coadjoint orbit is a generalised flag manifold 18 , a particular form of algebraic varieties. On the other hand, compact Hermitian symmetric spaces are well-known to be classified. One class of manifolds lying at the intersection of these two are the complex Grassmanians, given by the quotient SU (n + m)/S(U (n) × U (m)) which includes as the m = 1 case the complex projective spaces CP n .
Being complex manifolds, they come with an integrable complex structure J. For a given choice of the parameter c, the mCYBE coincides with the condition of vanishing Nijenhuis tensor: this observation provides an intuitive hint to the fact that the R-matrix on CP n , thought of as the coset space above, could be interpreted as a complex structure. This statement has been made precise by Koszul [63] who has proved that, in order for R to be a complex structure (in flat indices) on a reductive coset m (where M = G/H), it is required to satisfy, besides the mCYBE (now interpreted as an integrability condition), the constraints In addition, analogously to the Darboux theorem for standard complex structures, the R-matrix can be diagonalized [64]: if we consider the two subalgebras q,q such that g C = q ⊕q and q ∩q = h C then it can be shown that the conditions of the Koszul theorem are met provided that