Poisson-Lie duals of the η-deformed AdS2 × S2 × T6 superstring

We investigate Poisson-Lie duals of the η-deformed AdS2 ×S2 ×T6 superstring. The η-deformed background satisfies a generalisation of the type II supergravity equations. We discuss three Poisson-Lie duals, with respect to (i) the full psu1,12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{p}\mathfrak{s}\mathfrak{u}\left(1,\left.1\right|2\right) $$\end{document} superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard type II supergravity equations. The metrics and B-fields for the first two cases are the same and given by an analytic continuation of the λ-deformed model on AdS2 × S2 × T6 with the torus undeformed. However, the RR fluxes and dilaton will differ. Focusing on the second case we explicitly derive the background and show agreement with an analytic continuation of a known embedding of the λ-deformed model on AdS2 × S2 in type II supergravity.

To define a particular η-deformation of the semi-symmetric space sigma model, we first need to specify an antisymmetric operator R satisfying the non-split modified classical Yang-Baxter equation on the superalgebra psu(1, 1|2). We will take this R-matrix to be given by the canonical Drinfel'd-Jimbo solution associated to a particular Dynkin diagram and Cartan-Weyl basis of the superalgebra. 1 For such a choice of R-matrix the manifest symmetry algebra of the deformed model is broken to the Cartan subalgebra. Together with the remaining charges, which are hidden, the isometry algebra is q-deformed [6,14,[22][23][24] with q ∈ R depending on the string tension and the deformation parameter η.
The model is constructed such that on integrating out the degrees of freedom associated to an appropriate Borel subalgebra (that correlates with the R-matrix) we recover the η-deformation of interest. Following the results of [1], for subalgebras g 0 of psu(1, 1|2) corresponding to sub-Dynkin diagrams we can construct subalgebras of the complexified double psl(2|2; C) whose associated degrees of freedom can be integrated out to give the Poisson-Lie dual of the η-deformed AdS 2 × S 2 × T 6 superstring with respect to g 0 . Any additional Cartan generators not covered by the sub-Dynkin diagram can also be included in g 0 . It is likely that this is not a complete list of possible Poisson-Lie duals of the η-deformed AdS 2 × S 2 × T 6 superstring (see, for example, [30]). In this paper we mostly work with the Dynkin diagram − ⊗ − . A discussion of the other possible Dynkin diagrams is given in appendix A. Let us briefly outline three possible Poisson-Lie duals that one can consider based on this choice: 1. First, one can consider the Poisson-Lie dual with respect to the full psu(1, 1|2) superalgebra. This model is conjectured [1,[31][32][33][34][35] to be an analytic continuation of the λ-deformation of the AdS 2 × S 2 × T 6 superstring [36] (generalising the bosonic λ-deformed models of [37,38]), which, following the terminology of [1] we refer to as the λ ⋆ -deformed model.

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2. Second, one can take the sub-Dynkin diagram formed of the two bosonic nodes. This corresponds to dualising with respect to the full bosonic subalgebra su(1, 1) ⊕ su (2). The bosonic part of this model coincides with the λ ⋆ -deformed model, however they differ in the fermionic part.
3. Finally, one can consider just the u(1) ⊕ u(1) subalgebra associated to the two Cartan generators. This model is conjectured to be equivalent to taking the two-fold T-dual of the η-deformed AdS 2 × S 2 × T 6 superstring.
There is substantial evidence [39][40][41][42][43][44][45][46] that a Weyl anomaly is associated to integrating out the degrees of freedom of a non-unimodular algebra, that is when the trace of the structure constants is non-vanishing, f ab b = 0. In this case, rather than solving the standard supergravity equations, the background solves a generalisation thereof [47,48] (as discussed in the context of non-abelian duality in [49][50][51]). These generalised supergravity equations are equivalent to the κ-symmetry of the Green-Schwarz superstring [48]. They are also related to the standard supergravity equations by T-dualising a supergravity background in a U(1) isometry, y → y + c, which is a symmetry of all the fields except the dilaton, Φ ∼ y + . . . [47]. The relation with dualities has been explored further in the context of generalised geometry, double field theory and exceptional field theory [30,[52][53][54][55][56].
The η-deformation of S 2 [6,24] is equivalent [57] to the sausage model of [58]. A proposal for the η-deformation of the AdS 2 × S 2 × T 6 supergravity background solving the generalised supergravity equations is given in appendix F of [47] (see also [59][60][61]). This is consistent as the Borel subalgebra, whose degrees of freedom we integrate out to give the η-deformed AdS 2 × S 2 × T 6 superstring, is not unimodular. For the three duals listed above the subalgebras whose degrees of freedom we integrate out are all unimodular, and hence the corresponding backgrounds are expected to solve the standard supergravity equations. Note that, since all three models involve dualising in a timelike direction, these solutions may actually be of type II or II ⋆ supergravity [67]. For the Poisson-Lie dual with respect to the full psu(1, 1|2) superalgebra, assuming the conjectured relation to the λ-deformation [31][32][33][34], this is indeed the case [62]. It is also true for the twofold T-dual [32,47,63]. For the remaining case, we recall that an alternative, arguably simpler, embedding of the metric of the λ-deformed model in supergravity to that of [62] is given in [64]. The analytic continuation of this background therefore provides a natural conjecture for the Poisson-Lie dual of the η-deformed AdS 2 × S 2 × T 6 superstring with respect to the full bosonic subalgebra. The main result of this paper is to confirm this proposal.
That the same metric and B-field can be supported by different RR fluxes is known in the literature. Indeed, it is the case for the different embeddings of the metric of the λ-deformed model on AdS 2 × S 2 [62,64] and is also discussed in [65] in the context of the η-deformed models. Here, one mechanism by which this may happen, that is duality transformations with respect to different subalgebras, is studied.
The layout of this paper is as follows. In section 2 we review the model on the Drinfel'd double and the formalism for constructing Poisson-Lie dual sigma models. We then focus on the η-deformation of the AdS 2 × S 2 supercoset in section 3, writing its action in a JHEP08(2018)107 manifestly Poisson-Lie symmetric form. Using these results, in section 4 the background of the Poisson-Lie dual with respect to the full bosonic subalgebra is derived. In appendix A we discuss the different Dynkin diagrams of psu(1, 1|2) in the context of η-deformations and Poisson-Lie duality. In appendix B and appendix C we give our conventions for the superalgebras psu(1, 1|2) and pb(1, 1|2) and 4-dimensional and 32-dimensional gamma matrices respectively, while appendix D contains technical details of the derivation of the background of the Poisson-Lie dual with respect to the full bosonic subalgebra.
2 Poisson-Lie duality and the Drinfel'd double The Drinfel'd double. Poisson-Lie duality [2,3] is a generalisation of non-abelian duality to certain backgrounds that do not necessarily possess manifest isometries. The underlying algebraic structure is a Drinfel'd double, defined as a 2n-dimensional real connected Lie group D whose Lie algebra d = Lie(D) can be decomposed as where g andg are two n-dimensional real Lie subalgebras, maximally isotropic with respect to a non-degenerate ad-invariant inner product β(·, ·) on d, When d, g andg are Lie superalgebras, the Z 2 grading allows one to decompose them as where d B and d F contain the elements of grade zero and one respectively. The inner product should also be consistent with the Z 2 grading, supersymmetric and ad-invariant where [·, ·} is the Z 2 -graded commutator.
First-order action on the Drinfel'd double. As for abelian and non-abelian duality, dual models can be obtained starting from an action for a dynamical field in the Drinfel'd double and integrating out half the degrees of freedom. To define this first-order action on the Drinfel'd double we extend the definition of the inner-product to the Grassmann envelope of the superalgebra as

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and so on, where θ 1 , θ 2 are real Grassmann variables and c ∈ C is such that |c| = 1 and c θ 1 θ 2 ∈ R. If c is a complex number with complex conjugate c ⋆ , then we take the conjugation operation on Grassmann variables to be given by and correspondingly The inner product between two elements of different grading vanishes. The first-order action for the dynamical field l ∈ D that gives the Poisson-Lie dual models is [25,26] S D (l) = dτ dσ is the standard Wess-Zumino term. The quadratic form K, whose explicit form is discussed below, is model dependent and acts on the current, which takes values in the Grassmann envelope of the algebra d. Henceforth, we will use d, g,g and so on to refer to both the algebra and its Grassmann envelope.
Canonical Poisson-Lie duality. Starting from the action (2.9) and using the decomposition of the Drinfel'd double (2.1), where we recall both g andg are subalgebras, we recover the Poisson-Lie dual models on G andG by integrating out the degrees of freedom associated tog and g respectively [25,26]. To obtain the explicit form of the dual models we need to specify the action of the bilinear form K on an arbitrary element x ∈ d. Such an element admits a unique decomposition x = y + z where y ∈ g and z ∈g. Without loss of generality one may then define the action of the bilinear form as where G 0 and B 0 are the symmetric and antisymmetric parts of some operator F 0 :g → g with respect to the inner product ·, · . To integrate out the degrees of freedom associated tog, we parametrise the field l ∈ D as l =gg, whereg ∈G and g ∈ G. Taking where P g (respectively Pg) takes an element of the Drinfel'd double and projects it onto g (respectivelyg). The operator Pg Ad −1 g Pg is invertible ong and hence it is possible to eliminate y in favour of z,

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The action then becomes quadratic in z, which can be integrated out to give where the worldsheet light-cone coordinates are defined as To obtain the dual model, we parametrise the field l ∈ D as l = gg and integrate out the degrees of freedom associated to g. After exchanging the role of g andg in the above derivation one obtains the dual sigma model (2.16) The two sigma models (2.14) and (2.16) are described by the same set of equations after appropriate non-local field and parameter redefinitions. The model corresponding to the action (2.16) is said to be the dual of (2.14) with respect to g. Non-abelian duality is a special case of Poisson-Lie duality, and corresponds to the case in which the dual algebrã g is abelian and hence Π(g) = 0.
Poisson-Lie duality with respect to a subalgebra. Besides the canonical decomposition d = g ⊕g of (2.1), it is also possible to consider more general maximally isotropic decompositions of the Drinfel'd double of the type d = k ⊕k where k is not necessarily an algebra [4,25,26,34]. On the other hand,k is still a subalgebra of d and its associated degrees of freedom can be integrated out to yield a model on the coset spaceK\D. Starting with the field l ∈ D parametrised as l =kk, wherek ∈K = exp[k] and k ∈K\D, and integrating out the degrees of freedom associated tok following the steps outlined above, we find the following Lorentz-invariant action for the field k where Π(k) = P k Ad −1 k Pk(Pk Ad −1 k Pk) −1 , P rot k = P k − Π(k)Pk , P rot k = Pk + Π(k)Pk . (2.18) The operator F ′ 0 :k → k is related to F 0 :g → g via the non-linear transformation In the particular case where the intersection ofk with g defines a common subalgebra g 0 and one has the decomposition [1,30] g = g 0 ⊕ m ,g =g 0 ⊕m , we can interpret (2.17) as the Poisson-Lie dual of the model on G with respect to g 0 . The requirement thatk forms an algebra imposes restrictions with respect to which subalgebras it is possible to dualise.

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3 The η-deformed AdS 2 × S 2 supercoset In this section we turn our attention to the η-deformed AdS 2 × S 2 × T 6 superstring and its Poisson-Lie duals. To this end we consider the η-deformed semi-symmetric space sigma model [9][10][11][12][13][14] on the supercoset PSU(1, 1|2) SO(1, 2) × SO (2) , which we will refer to as the η-deformed AdS 2 × S 2 supercoset model, and investigate Poisson-Lie duals with respect to various subalgebras of psu(1, 1|2). We start by explaining how to write the η-deformed semi-symmetric space sigma model in the manifestly Poisson-Lie symmetric form (2.14). For this we need to specify the Drinfel'd double together with its invariant inner product, as well as the specific form of the operator F 0 . We then specialise to the supercoset (3.1), discussing the possible Poisson-Lie duals of the η-deformed AdS 2 × S 2 supercoset model.

The η-deformed semi-symmetric space sigma model
The action of the deformed model. Let g be an element of a supergroup G whose corresponding Lie superalgebra g is basic and admits a Z 4 grading consistent with the commutation relations The Z 4 grading follows from the existence of a linear automorphism of the complexified superalgebra Ω : g C → g C satisfying The bosonic subalgebra of g is given by g (0) ⊕ g (2) , while the fermionic generators belong to either g (1) or g (3) . We introduce the projectors P k g = g (k) , k = 0, 1, 2, 3, and denote the group corresponding to the grade 0 subalgebra by H = G (0) = exp[g (0) ]. The η-deformed model describes a deformation of the semi-symmetric space sigma model on the supercoset G/H. Its action is [13] S η (g) = T d 2 σ STr g −1 ∂ + g, P 1 where the supertrace STr denotes an ad-invariant and Z 4 invariant bilinear form on g, which is symmetric (respectively antisymmetric) on the bosonic (respectively fermionic) subspace of g and hence it is symmetric on the Grassmann envelope. The operators P and R g are given by where the operator R satisfies the non-split modified classical Yang Baxter equation

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and is antisymmetric with respect to the supertrace STr(X, RY ) = − STr(RX, Y ). We will take this R-matrix to be given by the canonical Drinfel'd-Jimbo solution associated to a particular Dynkin diagram and Cartan-Weyl basis of g. The overall coupling constant T plays the role of the effective string tension and η is the deformation parameter, flipping the sign of which is equivalent a parity transformation. The global left-acting G symmetry is broken to the Cartan subgroup, while the right-acting gauge symmetry, g → gh, where h belongs to H, is preserved.
Poisson-Lie symmetric action. To write the action of the η-deformed semi-symmetric space sigma model in a manifestly Poisson-Lie symmetric form, we recall that for the ηdeformed models the relevant Drinfel'd double is the complexified Lie algebra d = g C [4,31], which as a real vector space admits the decomposition whereg is the Borel subalgebra, formed by the Cartan generators and the positive roots are the Cartan generators, positive and negative roots respectively, then the Borel subalgebra is spanned byg = {h i , e M , ie M }. Furthermore, the action of the Drinfel'd Jimbo R-matrix on the Cartan-Weyl basis is given by To specify the operator F 0 we introduce bases of g andg, denoting the generators of g (respectivelyg) by T A , A = 1, 2, . . . , dim G (respectivelyT A ). We further assume that we have an inner product ·, · on g C with respect to which the two subalgebras are isotropic, T A , T B = T A ,T B = 0 and that provides a canonical pairing, that is A for bosonic generators and θ 1 T A , θ 2T B = i θ 1 θ 2 δ B A for fermionic ones. Introducing κ AB = STr(T A T B ) and the auxiliary operator we define the operators F P :g → g and F R :g → g by and These operators preserve the Z 2 grading of the superalgebra. The action (3.4) of the η-deformation is then equivalent to which is of the form (2.14) with The complex Drinfel'd double. The isometry algebra of the AdS 2 ×S 2 supercoset (3.1) is g = psu(1, 1|2), the bosonic subalgebra of which is su(1, 1) ⊕ su(2). As discussed above, the relevant Drinfel'd double for the η-deformed model is the complexified Lie superalgebra d = g C = psl(2|2; C). The Drinfel'd double can be decomposed into two real subalgebras where pb(1, 1|2) is the projected Borel subalgebra spanned by the Cartan generators and positive roots of psl(2|2; C). It will often be convenient for us to work with the superalgebra sl(2|2; C). The superalgebra psl(2|2; C) is then obtained by quotienting out the u(1) ideal, that is we identify elements of sl(2|2; C) that differ by the central element.
At this point let us make a brief comment on the different Dynkin diagrams of sl(2|2; C). In general, for Lie superalgebras there are inequivalent choices for the set of Cartan generators and simple roots, where the latter can either be bosonic or fermionic. Since the Drinfel'd Jimbo R-matrix (3.8) is defined by its action on the Cartan generators and roots, it is possible that different choices of Dynkin diagrams and Cartan-Weyl bases define inequivalent deformations. The three Dynkin diagrams of sl(2|2; C) are − ⊗ − , ⊗ − − ⊗ and ⊗ − ⊗ − ⊗, where represents a bosonic root and ⊗ a fermionic root. In this paper we will focus on the distinguished Dynkin diagram − ⊗ − . A discussion of the other choices is given in appendix A.

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Poisson-Lie duals and unimodularity. The background of the η-deformed AdS 2 × S 2 × T 6 superstring has a non-vanishing, albeit closed, B-field, together with a three-form and five-form RR flux [47,61]. 2 This background does not solve the standard supergravity equations, rather a generalisation thereof [47,48]. In the context of the first-order action on the Drinfel'd double and duality, the corresponding Weyl anomaly is expected to be associated to integrating out the degrees of freedom of a non-unimodular algebra, that is when the trace of the structure constants is non-vanishing, f ab b = 0. Indeed, when starting from the first-order action on the Drinfel'd double, the η-deformed model is obtained by integrating out the degrees of freedom associated to the projected Borel algebra pb(1, 1|2), which is indeed non-unimodular.
As the background of the η-deformed AdS 2 × S 2 × T 6 superstring solves the generalised supergravity equations, the Weyl anomaly is of a particularly special type. As a result, the background of the η-deformed model is expected to be related to solutions of the standard supergravity equations by Poisson-Lie duality. These dual models are given by integrating out the degrees of freedom of unimodular algebras in the model on the Drinfel'd double. We finish this section by listing three examples of such Poisson-Lie duals of the η-deformed AdS 2 × S 2 supercoset and describe the associated unimodular subalgebras of psl(2|2; C).
1. First, it is possible to dualise with respect to the full superalgebra psu(1, 1|2). In this case one integrates out the degrees of freedom associated to psu(1, 1|2), a unimodular algebra. In the terminology of [1] this gives the λ ⋆ -deformed model and is conjectured to be an analytic continuation of the λ-deformation [1,[31][32][33][34][35]. The background of the λ-deformation of the AdS 2 × S 2 × T 6 superstring has been derived in [62]. It has a vanishing B-field and the metric is supported by a RR five-form flux and dilaton giving a solution of the standard supergravity equations.
2. Second, as we are considering the distinguished Dynkin diagram − ⊗ − , the results of [1] tell us that we can dualise with respect to the full bosonic subalgebra of psu(1, 1|2), that is su(1, 1) ⊕ su (2), by considering the sub-Dynkin diagram formed of the two bosonic nodes. In this case the degrees of freedom that are integrated out are associated to the algebrak = su(1, 1) ⊕ su(2) ⊕ {Q Iαα }, where {Q Iαα } are the dual fermionic generators, that is the positive fermionic roots. Since this is also a unimodular algebra we expect the resulting background to again solve the standard supergravity equations.
The bosonic part of this model, and hence the metric and B-field, coincides with the λ ⋆ -deformation. However, they differ in the fermionic part. In [64] an alternative embedding of the metric of the λ-deformed AdS 2 ×S 2 ×T 6 superstring in supergravity was given. The metric is again supported by a RR five-form flux and dilaton, however these are different to those found in [62]. In section 4 we show that this background corresponds to an analytic continuation of the Poisson-Lie dual of the η-deformed AdS 2 × S 2 × T 6 superstring with respect to the full bosonic subalgebra.

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3. Finally, we consider the two-fold T-dual of the η-deformed AdS 2 × S 2 supercoset, equivalent to dualising with respect to the u(1)⊕u(1) Cartan subalgebra of psu(1, 1|2). As discussed in detail for the bosonic case in [1], one can show that the algebra whose degrees of freedom are integrated out is unimodular. Accordingly, the background of the two-fold T-dual of the η-deformed AdS 2 × S 2 × T 6 superstring solves the standard supergravity equations, again supported by a RR five-form flux and dilaton [32,47,63].
As shown in [32] and [62] respectively, the two-fold T-dual can be found by analytically continuing and taking a scaling limit of the backgrounds of [64] and [62]. The analytic continuation amounts to considering the reality conditions relevant for the η-deformed models, and hence the two-fold T-dual should be given by a real scaling limit of the two Poisson-Lie duals of the η-deformed AdS 2 × S 2 × T 6 superstring discussed above.
These three examples of Poisson-Lie duals of the η-deformed AdS 2 × S 2 × T 6 superstring all involve dualising in a timelike direction. Abelian T-duality in a timelike direction maps solutions of type II supergravity to solutions of type II ⋆ [67]. As Poisson-Lie duality is a generalisation of abelian T-duality, the corresponding backgrounds are expected to solve the standard type II ⋆ supergravity equations.

Poisson-Lie duality with respect to the full bosonic subalgebra
In this section we derive the background of the Poisson-Lie dual of the η-deformed AdS 2 × S 2 × T 6 superstring with respect to the full bosonic subalgebra su(1, 1) ⊕ su(2). We start from the first-order action on the Drinfel'd double (2.9) with F 0 = F P + F R defined in eqs. (3.10) and (3.11). We then consider the decomposition (2.20) with and integrate out the degrees of freedom associated to the algebrak = g 0 ⊕m, wherem is spanned by the positive fermionic roots. These degrees of freedom are associated to a unimodular algebra and hence the corresponding background is expected to solve the standard supergravity equations. Expanding the action (2.17) to quadratic order in fermions, we rewrite it in Green-Schwarz form and extract the background fields. The resulting background indeed solves the standard supergravity equations and, as conjectured, is given by an analytic continuation of that constructed in [64].

Parametrisation
In order to Poisson-Lie dualise with respect to the full bosonic subalgebra g B = su(1, 1) ⊕ su(2) we need to find a suitable parametrisation of the field k ∈K\D/H appearing in the action. Starting with a group-valued field k ∈ D = PSL(2|2; C) and using the fermionic part of the left-actingK gauge symmetry we partially gauge fix To gauge fixg 0 we writeg 0 =ǧ 0 ⊕ĝ 0 , whereǧ 0 corresponds to the AdS 2 factor andĝ 0 to the S 2 factor and gauge fix in each sector separately.
Gauge fixing in the S 2 sector. Let us introduce the generators S A andS A , A = 4, 5, 6, defined in terms of the Cartan generator h = σ 3 and the simple roots e = σ + , f = σ − of sl(2; C) as The right-acting gauge symmetry is generated by S 4 , the adjoint action of which rotatesS 5 andS 6 amongst themselves. Therefore, using this right-acting gauge symmetry together with the left-acting gauge symmetry generated by {S 4 , S 5 , S 6 }, we can partially gauge fix where the residual left-acting gauge symmetry is generated by S 5 . Using this residual gauge freedom we choose the familiar parametrisation of this coset which in terms of the generators {P 2 ,P 3 ,J 23 } defined in appendix B is given bŷ Gauge fixing in the AdS 2 sector. For the AdS 2 sector we introduce the following generators {S A ,S A }, A = 1, 2, 3, of sl(2; C) Here the right-acting gauge symmetry is generated by S 1 , the adjoint action of which hyperbolically rotatesS 2 andS 3 amongst themselves, while the left-acting gauge symmetry is generated by {S 1 , S 2 , S 3 }. Following the same logic as for the S 2 sector we find it is possible to gauge fixǧ which in terms of the generators {P 0 ,P 1 ,J 01 } defined in appendix B is given by Full parametrisation. Finally, our parametrisation of the field k ∈K\D/H is given by JHEP08(2018)107

NSNS background
The NSNS fields (the metric and the B-field) are obtained by setting the fermions in the action (2.17) to zero and considering the bosonic Lagrangian 3 which corresponds to the Lagrangian of the λ ⋆ -deformation of the bosonic sigma model on AdS 2 × S 2 and we have introduced the deformation parameter [24] κ = 2η 1 − η 2 . (4.14) Using the parametrisation (4.12) we find the following metric The B-field vanishes. After the coordinate redefinition 4 the metric becomes conformally flat, We also introduce the vielbein satisfying G M N = E M a E N b η ab . The spin connection, which we will need to obtain the RR fields, is given in terms of the vielbein 3 In our conventions the bosonic part of the action is related to the metric by and the line element is ds 2 = GMN dX M dX N . That is the tension T is included in the metric. 4 The redefinitions φ1 = log |p + p 2 +q 2 − 1| and φ4 = log |p + p 2 −q 2 − 1| give the same result.

RR background
To obtain the RR fluxes we rewrite the deformed AdS 2 × S 2 supercoset sigma model in Green-Schwarz form. The supercoset sigma model has 4 bosonic and 8 fermionic fields compared to the 10 bosonic and 32 fermionic of the type II Green-Schwarz superstring sigma model. In this subsection we describe a consistent truncation of the latter that can be matched with the former to quadratic order in fermions. This in turn allows us to derive the RR fluxes of the Poisson-Lie dual of the η-deformed AdS 2 × S 2 × T 6 superstring with respect to the full bosonic subalgebra su(1, 1) ⊕ su(2).

Truncation of the Green-Schwarz action
To embed the 4 dimensions of the λ ⋆ -deformed model on AdS 2 ×S 2 (4.17) into 10 dimensions we take the remaining six dimensions to simply be a flat torus, T 6 , so that the full 10dimensional metric is (i = 4, 5, 6,7,8,9) (4.26) For the truncation of the fermionic fields [19,62] let us start with the action of the type IIB Green-Schwarz superstring at quadratic order in fermions 5 where The action is written in conformal gauge with α, β = +, − and the worldsheet light-cone coordinates defined in eq. (2.15). The (rescaled) conformal gauge metric and two-index antisymmetric tensor are given by σ +− 1 = σ −+ 1 = ǫ +− = −ǫ −+ = 1. E αA and ω αAB are the pullbacks of the vielbein and corresponding spin connection to the worldsheet, H is the field strength of the B-field, Φ is the dilaton and F A , F ABC and F ABCDE are the RR fluxes.
It is useful to split the tangent space index A = 0, 1, 2, . . . , 9 into an index a = 0, 1, 2, 3 covering the AdS 2 × S 2 directions and an index i = 4, 5, 6, 7, 8, 9 covering the torus directions. As we are considering the type IIB superstring, the two 32-components spinors Θ I , I = 1, 2 are both Weyl spinors of the same chirality (Γ 11 Θ 1 = Γ 11 Θ 2 ) and satisfy the Majorana condition 6Θ The deformed supercoset sigma model has 8 fermionic fields compared to the 32 of the Green-Schwarz action. In order to rewrite the action in Green-Schwarz form we embed two 4-component fermionsθ I into the two 32-component spinors (4.31) 5 Since we are primarily interested in extracting the RR background fields we only consider the action of the Green-Schwarz superstring up to quadratic order in the fermions. Note that the overall factor is proportional to √ T as the metric (4.17) and vielbein (4.18) contain a factor of T and √ T respectively. 6 See appendix C for the definitions and properties of the gamma matrices and the charge conjugation matrix C.

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where the complex numbers a (I) , b (I) , c (I) and d (I) may take different values depending on the grading. Note that for our choice of gamma matrices the 32-component spinors are of negative chirality, Γ 11 Θ I = −Θ I . A priori, there is a non-trivial coupling between the spinors and the torus directions giving rise to terms such asΘ∂X i Θ. Such terms should not survive the truncation since they do not appear in the deformed supercoset sigma model. Requiring that this is indeed the case constrains the complex numbers a (I) , b (I) , c (I) and d (I) .
To find the complex numbers we assume that the deformed model is supported only by a self-dual RR five-form flux of the form where Ω 3 = ǫĩkdZĩ ∧dZ ∧dZk is the holomorphic three-form on the torus, 7 F 2 = 1 2 F ab E a ∧ E b and ⋆ is the 10-dimensional Hodge dual 8 squaring to one, ⋆ 2 = +1. With this ansatz where is a projector, (P 4 ) 2 = P 4 , that can be used to decompose the 32-component spinors The additional properties setting Θ I ⊥ = 0 imposes conditions on the complex numbers a (I) , b (I) , c (I) and d (I) . In particular, these are satisfied when a (I) = d (I) = 0, c (I) = −b (I) . Henceforth, we will consider this truncation and take the 32-component spinors and their Dirac conjugates to be The complex coordinates are chosen to be Z 1 = X 4 + iX 5 , Z 2 = X 6 + iX 7 and Z 3 = X 8 + iX 9 . 8 The Hodge dual is defined such that with ǫ 0123456789 = +1.

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Using the 32-dimensional gamma matrices given in appendix C we havē 39) and the Lagrangian of the Green-Schwarz action can be rewritten (4.40) Finally, let us comment on the reality condition satisfied by the 4-dimensional fermionŝ θ I . The Majorana condition (4.30) implies The two fermions can thus have different reality conditions depending on the choice of the complex numbers b (I) . We will choose to work with such that the Majorana condition becomes (4.43)

Field redefinitions
To match the form of the Green-Schwarz Lagrangian (4.40) we start by parametrising the field k of the deformed supercoset sigma model as in eq. (4.11) and expand the action (2.17) up to quadratic order in the fermions. We also use the Polyakov-Wiegmann identity to reduce the Wess-Zumino term to a two-dimensional integral. The resulting Lagrangian consists of four distinct parts: L 0 ∼ ∂X∂X is the bosonic Lagrangian giving rise to the metric of the λ ⋆ -deformed model and is discussed in subsection 4.2, L ∂ ∼ ∂Xθ∂θ contains the terms with one derivative acting on the fermions, L m ∼ ∂X∂Xθθ are the fermion "mass" terms and finally L ∂∂ ∼ ∂θ∂θ. Field redefinitions are needed to rewrite L ∂ , L m and L ∂∂ in Green-Schwarz form, in particular matching the consistent truncation (4.40). In appendix D we show that L ∂∂ is a total derivative and thus can be ignored. We then focus on two types of transformations, namely shifts of the bosons X → X + θs(X)θ and rotations of the fermions θ → r(X)θ, which lead to the following modifications

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Therefore, we would like to find functions s(X) and r(X) such that where the equalities hold up to total derivatives. In order to find the exact functions s(X) and r(X) satisfying these conditions we follow the procedure outlined in [66]. All terms quadratic in the fermions contributing to the Lagrangian can be classified according to their symmetry properties under the exchange of the two fermions. At quadratic order in fermions the Lagrangian can be written as L = L + + L − , where L + and L − contain the terms with the symmetry property In the above expression the sum over the spinor indices is understood: f IJ ± are 4×4 matrices depending on the bosons. In particular, this decomposition can be applied to the terms containing one derivative acting on the fermions, L ∂ = L ∂ + + L ∂ − . In the Green-Schwarz Lagrangian we have thatL ∂ GS + = 0, that is the Lagrangian only contains terms with the symmetry property Having identified the contributions to L ∂ ± we use a field redefinition to set L ∂ + equal to zero. Observing that rotating the fermions does not affect the symmetry property, this can only be achieved with a shift of the bosons and fixes for us the function s(X). Finally, the rotation of the fermions is implemented to rewrite the remaining terms in L ∂ in Green-Schwarz form. More details, including the exact field redefinitions r(X) and s(X), are presented in appendix D. After using the field redefinitions we find that the conditions (4.46) are satisfied, the Lagrangian takes precisely the form of the Green-Schwarz Lagrangian (4.40) and we can easily read off the RR fluxes, or more precisely the combination e Φ F .

RR fluxes
Comparing with the form of the Green-Schwarz Lagrangian (4.40) we find that the only necessary components of the two-form F 2 in eq. (4.32) are which leads to the following RR five-form flux This flux is imaginary, a consequence of the fact that we have dualised in a timelike direction and are now strictly speaking in type IIB ⋆ supergravity [67]. Since the two spinors Θ I

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satisfy the Majorana condition, this implies that the corresponding action of the Green-Schwarz superstring is not real. However, the Poisson-Lie dual of the η-deformed AdS 2 ×S 2 supercoset has a real action. This discrepancy comes from the rotation of the fermions θ I = U IJθJ , where θ I are the real 4-dimensional fermions appearing in the deformed supercoset model via the parametrisation (4.11), U IJ is the rotation matrix used to rewrite the action in Green-Schwarz form and the 4-dimensional fermionsθ I are related to the 32-components spinors Θ I as in eq. (4.31). When choosing the particular coefficients (4.42) the Majorana condition implies the reality conditions (4.43) for the 4-dimensional fermionsθ I :θ 1 is imaginary andθ 2 is real. However, the rotation matrix U IJ defined in eq. (D.20) has only real entries and thus we identify a real and an imaginary fermion, thereby breaking the reality condition. Implementing the transformation rules (4.21) and (4.24) we find which is precisely the RR five-form flux supporting the metric of the λ-deformed model on AdS 2 × S 2 × T 6 found in [64].

Concluding comments
In this paper we have investigated Poisson-Lie duals of the η-deformed AdS 2 × S 2 × T 6 superstring. While the background of the η-deformed AdS 2 × S 2 × T 6 superstring satisfies a generalisation of the standard supergravity equations, here we have discussed three Poisson-Lie duals, with respect to (i) the full psu(1, 1|2) superalgebra, (ii) the full bosonic subalgebra and (iii) the Cartan subalgebra, for which the corresponding backgrounds are expected to satisfy the standard supergravity equations. Focusing on the second case we explicitly derived the background, showing agreement with the embedding of the metric of the λ-deformed model on AdS 2 × S 2 in supergravity given in [64] up to an analytic continuation.
There are various interesting open questions extending and developing the results presented here. Firstly, it would also be useful to confirm some of the remaining conjectures summarised in this paper and extend the results to other backgrounds. This includes explicitly checking that Poisson-Lie dualising with respect to the full psu(1, 1|2) superalgebra gives the background of [62] up to analytic continuation and dualising with respect to the Cartan subalgebra gives the two-fold T-dual of [32,47,63]. Furthermore, candidates for the background of the Poisson-Lie dual of the η-deformed AdS 3 × S 3 × T 4 and AdS 5 × S 5 superstrings with respect to the full bosonic subalgebras are given in [64] and [68] respectively, up to analytic continuation.

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It is highly probable that the backgrounds discussed in this paper define integrable 2-dimensional sigma models. The results of [69] together with the integrability of the η-deformed principal chiral, symmetric space sigma and semi-symmetric space sigma models [5,6,13] suggest that their Poisson-Lie duals are also integrable. It is important to point out that the semi-symmetric space sigma model on the supercoset (1.1) is a truncation of the type II Green-Schwarz superstring on certain AdS 2 × S 2 × T 6 backgrounds. The integrability of the latter was demonstrated to quadratic order in fermions in [70]. In principle, for a complete analysis of integrability, this analysis should be extended to the deformed backgrounds.
As well as exploring deformations of other AdS 2 integrable string backgrounds [12,71], it would be interesting to consider some of the ways integrability has been used to study the η-and λ-deformed AdS 5 × S 5 superstrings in the context of the η-deformed AdS 2 × S 2 × T 6 superstring and its Poisson-Lie duals. This includes the light-cone gauge Smatrix [24,66,72,73], which should be a deformation of the S-matrix of [74], and the associated finite-size spectrum [75][76][77][78]. Another direction would be to investigate how the analysis of the solitons in the λ-deformed AdS 5 × S 5 superstring [79] is modified when considering the Poisson-Lie dual of the η-deformation with respect to either the full superalgebra (that is the λ ⋆ -deformed model, an analytic continuation of the λ-deformed model) or the full bosonic subalgebra. This is of relevance to both the AdS 5 × S 5 and AdS 2 × S 2 × T 6 cases. One could also ask how the q-deformed symmetry of the η-deformed model [6,14,[22][23][24] and the contraction limits (that is the maximal deformation, η → 1, limit) of [77,80,81] behave under Poisson-Lie duality.

Acknowledgments
We thank A. Tseytlin for comments on the draft. This work is supported by grant no. 615203 from the European Research Council under the FP7.

A Dynkin diagrams of sl(2|2; C)
The Lie superalgebra sl(2|2; C) admits three Dynkin diagrams where denotes a bosonic root and ⊗ a fermionic root. In this appendix we present the sl(2|2; C) superalgebra and for each Dynkin diagram give a Cartan-Weyl basis, the corresponding Cartan matrix and discuss the unimodularity properties of the Borel subalgebra spanned by the Cartan generators and positive roots.

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The sl(2|2; C) superalgebra. The bosonic subalgebra of sl(2|2; C) is sl(2; C)⊕sl(2; C)⊕ gl(1; C) for which we introduce the corresponding generators K 0 , K ± , L 0 , L ± and C 0 . We also introduce the eight supercharges Q ±αα whereα = ± is the spinor index associated to the first copy of sl(2; C) andα = ± to the second. The first index corresponds to the splitting of the supercharges under the gl(1; C) outer automorphism generated by R The non-vanishing commutation relations are 3) while the non-vanishing anticommutation relations for the supercharges read The Borel subalgebra is generated by Dynkin diagrams.

1.
− ⊗ − In this case there are two bosonic simple roots and one fermionic. A choice of Cartan generators and positive and negative simple roots is

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with the corresponding symmetrised Cartan matrix given by The non-simple roots are e 12 = −Q +−− , e 23 = −Q +++ , e 123 = +Q +−+ , The Borel subalgebra (A.7) is non-unimodular In this case there are two fermionic simple roots and one bosonic. The bosonic root can belong either to the first or second copy of sl(2; C). Although the two choices are symmetric, we shall present both for convenience.
• When the bosonic simple root comes from the first copy of sl(2; C) a choice for Cartan generators and positive and negative simple roots is with the corresponding symmetrised Cartan matrix given by The non-simple roots are e 12 = +Q +++ , e 23 = −Q −++ , e 123 = +L + , The Borel subalgebra (A.7) is non-unimodular • When the bosonic simple root comes from the second copy of sl(2; C) a choice for Cartan generators and positive and negative simple roots is

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with the corresponding symmetrised Cartan matrix given by The non-simple roots are e 12 = −Q −++ , e 23 = +Q +++ , e 123 = +K + , The Borel subalgebra (A.7) is non-unimodular 3. ⊗ − ⊗ − ⊗ In this case all three simple roots are fermionic. A choice of Cartan generators and positive and negative simple roots is with the corresponding symmetrised Cartan matrix given by The non-simple roots are e 12 = −K + , e 23 = +L + , e 123 = −Q +++ , The Borel subalgebra (A.7) is non-unimodular The η-deformed models and their Poisson-Lie duals. We conclude this appendix with a few comments on the η-deformed models that correspond to the different Drinfel'd-Jimbo R-matrices associated to the various Cartan-Weyl bases discussed above, together with their Poisson-Lie duals. The question of whether these η-deformations are inequivalent or not has not been previously studied. However, the Borel subalgebra (A.7) is nonunimodular in all three cases and thus we expect the corresponding η-deformed models to each have a Weyl anomaly. The backgrounds of the Poisson-Lie duals with respect to the full psu(1, 1|2) superalgebra and the Cartan subalgebra, which can be considered in all three cases, should solve JHEP08(2018)107 the supergravity equations as the degrees of freedom that are integrated out are associated to unimodular algebras. As discussed in [1] one can consider Poisson-Lie duals with respect to subalgebras that correspond to sub-Dynkin diagrams. The bosonic subalgebra su(1, 1) ⊕ su(2) corresponds to a sub-Dynkin diagram for choice 1, but not for choices 2 and 3. However, it is worth noting that it should still be possible to Poisson-Lie dualise the η-deformed models corresponding to choices 2 and 3 with respect to the full bosonic subalgebra by rewriting in terms of the Cartan-Weyl basis of choice 1. The same logic holds for any sub-Dynkin diagram of any of the three Dynkin diagrams.

B.2 Generators of pb(1, 1|2)
The projected Borel subalgebra pb(1, 1|2) is spanned by the Cartan generators and the positive roots. For the Dynkin diagram −⊗− with the matrix realisation of psu(1, 1|2) given above, these can be chosen to be upper triangular matrices. The duals of the psu(1, 1|2) generators can then be identified using the inner product (3.14). 9 For the definitions of the gamma matrices refer to appendix C.

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Bosonic generators. The six bosonic generators of pb(1, 1|2) are given bỹ Note that the fermionic supercharges anticommute among themselves and the central element does not appear in the commutation relations.

B.3 Mixed commutation relations
The Poisson-Lie dual of the η-deformed AdS 2 × S 2 supercoset with respect to the full bosonic subalgebra su(1, 1) ⊕ su(2) discussed in section 4 follows from integrating out the degrees of freedom associated to the algebrã k = {J 01 , J 23 , P a ,Q Iαα } . (B.14) To expand the action to quadratic order in fermions the following commutation relations are useful [θ I Q I ,P a ] = δ IJQJ γ 0 γ a θ I + 1 2 δ IJ Q Jγ a θ I , Using the Pauli matrices
(C.5) Furthermore, Dirac conjugation acts on 32-component spinors asΘ I = Θ † Γ 0 and the Majorana condition isΘ where the charge conjugation is defined as

D Field redefinitions
In this appendix we present the field redefinitions that bring the Poisson-Lie dual of the ηdeformed AdS 2 × S 2 supercoset with respect to the full bosonic subalgebra su(1, 1) ⊕ su(2) to Green-Schwarz form. Following the same notation as in subsection 4.3 we split the

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Lagrangian into four distinct parts, L = L 0 + L ∂ + L m + L ∂∂ . The terms quadratic in the fermions can also be split according to their symmetry properties under the exchange of the two fermions, L ∂ = L ∂ + + L ∂ − . To match the form of the Green-Schwarz Lagrangian we first show that L ∂∂ is a total derivative and thus can be ignored. We then identify L ∂ ± and find the appropriate shift of the bosons cancelling the terms with the wrong symmetry property, that is so that L ∂ + + (δL 0 ) ∂ + = 0. Finally, we rewrite the remaining terms, L ∂ − + (δL 0 ) ∂ − , in Green-Schwarz form with a rotation of the fermions.
Contribution to L ∂∂ . Parametrising the field of the deformed supercoset sigma model as in eq. (4.11) k =g 0 e χ ,g 0 ∈G 0 , we expand the action (2.17) to quadratic order in χ and find On an element of the Grassmann envelope the operator PmF −1 0 P m acts as and is thus antisymmetric with respect to the inner product. Moreover, since it does not depend on the bosons, L ∂∂ can be rewritten where α, β = +, − and ǫ +− = −ǫ −+ = 1, thus showing that L ∂∂ is a total derivative.
(D.5) To bring this expression closer to Green-Schwarz form we rewrite it in terms of the fermions θ I . We start by expanding the bosonic currents 10 g −1 0 ∂ αg0 = e αaP a + e α01J 01 + e α23J 23 ≡ e αAT A , (D. 6) where we have introduced e αA = e M A ∂ α X M and e αAB = e M AB ∂ α X M . The bosonic coordinates X M parametrise the group elementg 0 . We also define the operator
Rotation of the fermions. The final task is then to bring the remaining terms in L ∂ to Green-Schwarz form where we have used the Majorana condition (4.41). To proceed we compare the terms proportional to σ αβ 1 and ǫ αβ . Adding and subtracting the two resulting equations leads to T 2 e αA (G + B) Aa θ I γ 0 σ IJ 3 − 1 + η 2 1 − η 2 δ IJ + 2η 1 − η 2 σ IJ 1 γ a ∂ β θ J = 2(b (1) ) 2 E αaθ 1 γ 0 γ a ∂ βθ 1 , (D.19) where we note that onlyθ 1 orθ 2 appear on the right-hand side. To match this structure on the left-hand side we perform the following rotation of the fermions where U (J) , J = 1, 2, are two 4 × 4 matrices that are to be determined. This redefinition may lead to new "mass" terms ∼θ K U IK ∂ β U JLθL as the rotation matrices may in principle depend on the bosons. We will see that this is not the case here: the two rotation matrices U (J) do not depend on the bosons. DefiningŪ (I) = γ 0 U t (I) γ 0 and implementing the JHEP08(2018)107 Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.