Generalized Dualities and Supergroups

Using a recently developed formulation of double field theory in superspace, the graviton, $B$-field, gravitini, dilatini, and Ramond-Ramond bispinor are encoded in a single generalized supervielbein. Duality transformations are encoded as orthosymplectic transformations, extending the bosonic $O(D,D)$ duality group, and these act on all constituents of the supervielbein in an easily computable way. We first review conventional non-abelian T-duality in the Green-Schwarz superstring and describe the dual geometries in the language of double superspace. Since dualities are related to super-Killing vectors, this includes as special cases both abelian and non-abelian fermionic T-duality. We then extend this approach to include Poisson-Lie T-duality and its generalizations, including the generalized coset construction recently discussed in arXiv:1912.11036. As an application, we construct the supergeometries associated with the integrable $\lambda$ and $\eta$ deformations of the $AdS_5 \times S^5$ superstring. The deformation parameters $\lambda$ and $\eta$ are identified with the possible one-parameter embeddings of the supergravity frame within the doubled supergeometry. In this framework, the Ramond-Ramond bispinors are directly computable purely from the algebraic data of the supergroup.


Introduction
Abelian T-duality is an exact symmetry of perturbative string theory. Its initial formulation on an S 1 with associated isometries of metric and B-field can be straightforwardly extended to a d-dimensional torus, where the T-duality group expands to O(d, d; Z). Its modern description was given by Buscher [1,2], who couched it in the language of the effective worldsheet σ-models with commuting isometries; here one can derive the transformation rules of the metric and B-field by integrating out the worldsheet one-forms that gauge the isometries. Later work extended this approach to include the fermionic fields and the Ramond-Ramond sector from the target space perspective [3][4][5] and from the worldsheet using both the Green-Schwarz superstring [6,7] and the pure spinor superstring [8].
When these isometries no longer commute, it is no longer clear that the corresponding classical σ-model duality, known as non-abelian T-duality (NATD), is a full-fledged symmetry of string theory [9][10][11][12]. A symptom of this is that the dual space typically lacks local isometries that would permit one to invert the duality and recover the original space -the duality appears to be effectively one-way. Nevertheless, this procedure can still provide a means to systematically generate new supergravity solutions from existing ones.
Klimčík andŠevera showed that one can generalize the notion of duality, so that two or more σ-models related by NATD are indeed properly dual, in the sense that they can be derived from the same universal E-model [13][14][15]. In this framework, NATD is just the simplest example of Poisson-Lie T-duality (PLTD) [16,17], which can be further generalized to include a Wess-Zumino-Witten term [18] and a so-called dressing action [19], where one factors out local symmetries in very close analogy to the construction of σ-models on coset spaces. In this paper, we will be concerned with an even more general framework, known as a generalized coset [20]. The relations between these various concepts can be summarized as follows: abelian ⊂ non-abelian ⊂ Poisson-Lie ⊂ WZW-Poisson ⊂ ⊂ dressing coset ⊂ generalized coset .
One specifies a Lie group D with a split signature Killing metric η and a maximally isotropic subgroup H of half the dimension. In the absence of a dressing action, the physical space lies on the coset H\D, and in this context D is usually called a double Lie group. For the case of a generalized coset, there is an additional "dressing action" by another isotropic subgroup F , and the physical space is the double coset H\D/F . Different σ-models arise when there exist different choices for H, and these are related by this more general notion of duality.
In recent years, a modern perspective on these developments has been provided in the language of double field theory (DFT) [21][22][23][24]. 1 This is a generalization of supergravity incorporating T-duality manifestly in the target space geometry and low energy action of string theory. The coordinates x m of spacetime are "doubled" to include dual coordinates x m corresponding to winding modes of the string. The metric and B-field are combined into a generalized metric H. We decompose the coordinates and generalized metric as xm = (x m ,x m ) , Hmn = g mn − b mk g kl b ln b mk g kn −g mk b kn g mn . (1.1) In order to ensure that at most half the coordinates are physical, a section condition is imposed  [1,2]. In this sense, double field theory geometrizes T-duality. This bears a striking similarity to PLTD and indeed the two are intimately related [27], with PLTD and its generalizations corresponding to double field theory on group manifolds [28] or coset spaces [20]. This has been an active area of research in recent years (see e.g. [27][28][29][30][31][32][33] and references therein). As formulated in [21][22][23][24], DFT encompassed only the NS-NS sector (graviton, B-field, and dilaton). It has since been extended [34][35][36][37] to include the NS-R fermions (gravitini and dilatini) and the R-R sector (the even/odd p-form complexes) of type II string theory, but this extension did not fully unify the fields. The three sectors, NS-NS, NS-R, and R-R are encoded separately in the low energy type II action, and this complicates the construction of the dual supergravity backgrounds since one cannot address all sectors simultaneously using the same methods. Typically, one uses geometric or σ-model methods to fix some of the fields and then exploits the structure of κ-symmetry and supersymmetry to uncover the rest. The Ramond-Ramond sector is particularly onerous, since unlike the other bosonic fields, it does not appear explicitly in the Green-Schwarz σ-model action. 2 The goal of this paper is to address some of these topics from the perspective of a manifestly supersymmetric and duality covariant formulation. Such a formulation has recently been constructed by one of us in the language of double superspace [38], building off earlier work on the subject [26,[39][40][41][42]. Double superspace can be understood in a nutshell as simultaneously geometrizing supersymmetry and T-duality. In conventional superspace, the graviton (vielbein) and gravitino are unified into a single supervielbein, which in a certain gauge reads Diffeomorphisms and supersymmetry are unified into superdiffeomorphisms. In double superspace one is led to consider a generalized (double) supervielbein, which can be written in a certain gauge and duality frame as a product of three factors, The field E M A is the supervielbein, B M N is the super two-form (which appears in the Green-Schwarz action), and S BA includes "matter" fields, the dilatini and Ramond-Ramond bispinor. The duality group O(D, D), which governs the geometric structure of double field theory, is replaced by its natural supergroup analogue, the orthosymplectic group OSp(D, D|2s) with D bosonic coordinates, s fermionic coordinates, and their duals. Diffeomorphisms, B-field gauge transformations, and supersymmetry are all encoded in a single generalized superdiffeomorphism. Because all of the fields of supersymmetric double field theory are described in a single geometric object, one can apply the same techniques to derive how all of them transform under dualities, including abelian, non-abelian, and their generalized cousins.
A crucial point about conventional superspace is that it is not simply described by a super-Riemannian geometry with an unconstrained supermetric. Rather, one must employ the supervielbein and impose constraints on its torsion tensor in order to recover the physical field content of supergravity. These constraints involve θ-derivatives, but typically constrain the x-dependence as well, placing the geometry on-shell. In the Green-Schwarz superstring, these constraints arise from requiring κ-symmetry. Analogous statements hold for double superspace -we need to impose constraints on the generalized flux tensor F ABC in order for a supergravity interpretation to be possible, and these will coincide with the κ-symmetry constraints.
We begin in section 2 with a discussion of superspace double field theory, highlighting how the duality group OSp(D, D|2s) acts on the various constituents of V M A . These transformations provide the generic scaffolding in which all T-dualities must act. In section 3, as the simplest non-trivial example of such a transformation, we review the case of super non-abelian T-duality (NATD) in the Green-Schwarz superstring, where a supergroup G of isometries is dualized [43] (see [6,7,44] for earlier work on abelian T-duality of a single bosonic isometry and [45] for the non-abelian T-dual of supercoset σ-models). By comparing the dual Green-Schwarz models, one can deduce the form of the orthosymplectic transformation, which immediately yields the transformation rules of the supergravity fields, including the transformations of the Ramond-Ramond fields [46,47]. As a side benefit of this analysis, we are able to specialize to a fermionic isometry and recover results for fermionic T-dualities, both in the abelian [48][49][50][51][52] and non-abelian [53,54] cases. The case of non-abelian fermionic T-duality has been of particular interest recently, and we highlight the origin of the conditions given in [53,54] for the Killing spinor from the σ-model. 3 Non-abelian T-duality of the GS superstring provides a concrete example, exhibiting a number of important features that continue to hold for more general cases. In section 4, we introduce, following [20,28,59], the notion of a generalized parallelizable superspace, which is the natural analogue of a group manifold in the doubled setting, requiring only a double Lie group D and its maximally isotropic subgroup H. In section 5, we extend this framework to generalized supercosets, where an additional isotropic subgroup F is factored out, in direct analogy to the bosonic case [20]. In both of these discussions, we address two particular examples, D = G × G and D = G C , where G is a real super Lie group admitting an invertible Killing form. Both examples admit maximally isotropic subgroups H, the diagonal subgroup G diag for G × G and the real subgroup G for G C . The two groups G × G and G C can be analytically continued into each other, and the same holds true for their respective generalized geometries. For G C , another isotropic subgroup H is sometimes possible: it requires an R-matrix satisfying the modified classical Yang-Baxter equation. The two solutions for G C lead to backgrounds related by Poisson-Lie T-duality.
The discussion of generalized parallelizable superspaces and generalized supercosets in sections 4 and 5 is not really any different from their bosonic analogues: in effect, we simply insert a grading. In order to apply these results to supergravity, we must further impose additional κ-symmetry constraints on the generalized flux tensors. We review these in section 6 and discuss how they can be imposed in two specific cases: these are the so-called λ and η deformations of the AdS 5 × S 5 superstring. The λ deformation [60] (building off earlier work [61,62]) arises from a deformation of the non-abelian T-dual of the AdS 5 × S 5 superstring. The η deformation [63,64] is a type of Yang-Baxter σ-model [65,66] (see also [67]). Remarkably, these two different deformations preserve the integrable structure of the superstring, and this property has driven interest in them. From our perspective, these models are interesting because they can be very simply understood in the context of Poisson-Lie T-duality for the double Lie groups G × G and G C , respectively, where G is the superisometry group PSU(2, 2|4) of the AdS 5 × S 5 superstring. This interpretation was given in the language of E-models for the bosonic sector in [15]. Our main task in section 6 is to extend this to the fully supersymmetric case.
In addressing the λ and η models, we proceed ahistorically, and in fact, anti-chronologically. Beginning with the underlying double Lie structure of G×G and G C , we will seek to build a generalized supervielbein V M A whose flux tensor F ABC obeys the κ-symmetry constraints (6.1) and (6.2). For each case, there turns out to be a single one-parameter family, and this leads inexorably to the λ and η models upon identifying the underlying constituents of the Green-Schwarz action. All supergravity fields, including the Ramond-Ramond field strengths, are read directly off from the supervielbein and match the results derived by analyzing the respective Green-Schwarz σ-models [68].
In line with our ahistorical approach, we will not directly address issues of integrability or the connection between generalized duality and integrability. For a discussion of integrability, the reader is referred to the recent work [69], which explored some of these very issues for supersymmetric σ-models; specifically, it was shown that the Lax connection is preserved (a sufficient condition for integrability) after performing non-abelian T-duality in superspace analogues of the the principal chiral, symmetric space, and semi-symmetric space σ-models. On the connection between E-models and integrability, the reader is referred to [70,71].
We include several appendices. Our conventions for supergroups, including the orthosymplectic group, can be found in appendix A. We sketch some relevant results for type II supergravity in superspace in appendix B. A concise discussion of gauged superspace σmodels (whose results we employ in section 3) is given in appendix C. Finally in appendix D we give the generalized flux tensors for the η and λ models that are compatible with κ-symmetry.

Supersymmetric double field theory and the framework of T-duality
We will be employing the supersymmetric formulation of type II double field theory in superspace recently discussed in [38] (see also [39,40] and [41] for related earlier discussions). In this section, we will review some basic elements of this approach and explain how T-duality is manifested on the generalized supervielbein. As a first step, we will review some key features of bosonic double field theory, before showing how these generalize to the supersymmetric setting.

Bosonic double field theory and O(D, D) T-duality
Double field theory [21,24,26] is formulated on a space with local coordinates xm where fields are subject to a modified notion of generalized diffeomorphism governed by a Lie derivative L which preserves an O(D, D) structure. For vector fields Vm, where indices are raised and lowered with the constant O(D, D) metric ηmn. The space comes equipped with a generalized metric Hmn, which is an element of O(D, D) so that its inverse is (H −1 )mn = Hmn. Closure of generalized diffeomorphisms is guaranteed if we universally impose a section condition on all fields and parameters, ηmn∂m ⊗ ∂n = 0, where the derivatives may act either on the same or different fields. The metric and coordinates can be decomposed in terms of the GL(D) ⊂ O(D, D) subgroup as The section condition can then be solved by choosing∂ m = 0 universally. Then, the generalized metric H is described in terms of a metric g mn and a Kalb-Ramond two-form b mn as and the generalized Lie derivative decomposes into the conventional GL(D) Lie derivative and B-field transformations. The description in terms of a generalized metric turns out to not be particularly useful when passing to superspace. Just as supergravity requires that we exchange a metric g mn for a vielbein e m a , supersymmetric double field theory requires we replace the generalized metric Hmn with a generalized vielbein Vmâ. These are related by where The generalized vielbein can be decomposed as [25,72] which is the generic form if one supposes V a m and V a m to both be invertible matrices. This can be expressed as a product of two O(D, D) factors: The two vielbeins e m a and e m a describe the same metric, g mn = e m a e n b η ab = −ē m aē n b η ab implying that they are connected by a Lorentz transformation The double Lorentz symmetry can be fixed to a single Lorentz group by adopting the gauge Λ = 1. However, in supergravity this is more subtle because chiral fermions are present, breaking each Lorentz group to its connected (proper orthochronous) component. This means that Λ falls into one of four classes, depending on whether it preserves or reverses the temporal and spatial orientations: this distinguishes the type IIA/IIB/IIA * /IIB * duality frames [36,37,73]. Double field theory conveniently packages the O(D, D) structure of T-duality transformations. To see how, we define E nm := g nm −b nm andĒ nm := g nm +b nm = (E nm ) T = E mn . An O(D, D) transformation Unm acting on the right of Vâm can be written This recovers the Buscher rules for the metric and B-field and has the form of a fractional linear transformation on E nm . The fact thatĒ mn = E nm follows from the O(D, D) structure. Also encoded above is how the Lorentz transformation Λ a b that defines the type II duality frame is related to the original Λ a b . This can be written alternatively as a left or right Lorentz transformation Λ(U ), (2.13) Again, t+he fact that this is a Lorentz transformation follows from the O(D, D) structure. In addition to the generalized vielbein, double field theory also involves a generalized dilaton e −2d . This is a density under O(D, D) transformations, transforming as (2.14) Upon solving the section condition, the physical dilaton ϕ is identified by removing a density factor from the generalized dilaton, e −2d = e −2ϕ × det e m a . A generic transformation of the generalized dilaton is simply a scalar factor which is a priori independent of Umn. Together Umn and U ∆ encode an O(D, D) × R + transformation. It follows that the physical dilaton transforms as Note that detX m n = det X m n since X andX are related by a Lorentz transformation.

Supersymmetric type II double field theory
We turn now to supersymmetric type II double field theory [37]. At the component level, supersymmetric double field theory consists of the following fields: • the generalized vielbein Vmâ and the generalized dilaton e −2d ; • the gravitini Ψ aβ and Ψ a β , which are vectors and Weyl spinors under alternating Lorentz groups, and the dilatini ρ α and ρᾱ, which are Weyl spinors of opposite chirality to the gravitini; • and the Ramond/Ramond field strengths, which can be described equivalently as an O(D, D) spinor |F [34,35] or a Weyl bispinor [36,37].
In order to make contact with conventional superspace (and the Green-Schwarz superstring), a parametrization is needed that naturally leads to a supervielbein E M A and a Kalb-Ramond super-two-form B M N where z M = (x m , θ µ ) are the D bosonic and s fermionic coordinates of superspace. This can simply be done by mimicking the structure of bosonic double field theory, but replacing O(D, D) with its natural graded extension OSp(D, D|2s), the orthosymplectic supergroup involving 2D bosonic and 2s fermionic directions [41]. For type II superspace, we will need D = 10 and s = 32. For the details about this supergroup, we refer to appendix A.2. One begins by formulating supersymmetric double field theory on a superspace with local coordinates z M , with M a curved vector index of OSp(D, D|2s). Generalized diffeomorphisms act on a vector V M as where (−) m is −1 if M is fermionic and +1 otherwise. Indices are raised and lowered with the graded symmetric orthosymplectic invariant η MN subject to NW-SE rules, Closure of the gauge algebra is guaranteed by imposing the section condition η MN ∂ N ⊗ ∂ M = 0, exactly as in bosonic double field theory.
To recover conventional superspace, we decompose all objects carrying curved indices M under the GL(D|s) ⊂ OSp(D, D|2s) subgroup. The OSp(D, D|2s) metric in this basis is The coordinates and their derivatives decompose as where z M is the physical coordinate andz M is the winding coordinate. We normally solve the section condition by discarding any dependence on the winding coordinate. As in bosonic double field theory, we introduce a generalized supervielbein V M A with which to flatten generalized vectors. We choose it to be an OSp(D, D|2s) element, so that it is related to its inverse ( superspace, the flat index A decomposes in the chiral basis into two vector indices, one for each factor of the double Lorentz group, and four Weyl spinor indices, one of each chirality for each factor. We denote this for a vector V A as We have included above the relative dimension of these various components. These dimensions can be understood as arising from the R + factor in the decomposition OSp(10, This dimension is one reason why we should not combine the two 16-component Weyl spinors V α and V α into a single 32-component Dirac spinor.
We have normalized the relative dimension so that it leads to the correct notion of engineering dimension for the flat derivatives These matrices (and their chiral subblocks) are invariant under the double Lorentz group. As in the bosonic case, there are unphysical ingredients present in the supervielbein, which are associated with local symmetry transformations In the bosonic case, the local symmetry group is the double Lorentz group O(D − 1, 1) L × O(1, D −1) R with commuting left and right factors. In the supersymmetric case, this group is larger, although it still factors into two commuting chiral pieces. We denote it H L × H R . The generators λ AB of H L are constrained as in Table 1. Unlike the bosonic case, there is no simple prescription whereby some invariant H AB determines λ; instead, one needs to take into account the constraint structure on the supersymmetric worldsheet [26]. For further details of this symmetry group, we refer to [38,42].
There are competing ways of parametrizing a generic supervielbein, depending on whether one wishes to make contact with component double field theory or with type II superspace. In this paper, we will only be concerned with the latter. Then as shown in [38], a generic supervielbein can be decomposed as a product of three simple factors: The first is built out of the Kalb-Ramond super two-form, 25) just as in the bosonic case. The second factor V EΛ is written, in a chiral decomposition of the A index, as the relations (2.27). The V S factor is given, also in a chiral decomposition, as It consists of fermionic superfields S a β and S aβ as well as the symmetric bosonic superfields S αβ , S αβ , and S αβ . All these constituents transform as their indices imply under double Lorentz transformations, while only V S transforms under the higher dimension H L × H R transformations: The last condition implies that S αβ and S αβ are pure gauge, while the spin-1/2 parts of S a α and S aᾱ are invariant and constitute the dilatini The invariant components S αβ contain the Ramond-Ramond bispinor field strength (B.13). The precise dictionary between these constituents and those of type II supergravity are reviewed in Appendix B. It is instructive to compare the numbers of independent bosonic and fermionic components of these constituents with those of a generic OSp(D, D|2s) element. 4 Taking into account the range of the indices a = 0, . . . , D − 1 and α = 1, . . . , s/2, we count object bosonic fermionic From their embedding in double field theory, we will be able to derive the generic transformations of the supervielbein, dilatini, Ramond-Ramond sector, and dilaton under OSp(D, D|2s) transformations. For now, we will not concern ourselves with the precise form of these transformations. As we will discuss in the next sections, these encompass both bosonic T-duality [6][7][8]44] and fermionic T-duality [48] (see also [49][50][51][52]), as well as more general non-abelian dualities involving a supergroup G [43].
The key first step in uncovering the OSp structure is to introduce square matrices E A Note that whileĒ M a = E M b Λ b a , this does not hold for their inverses. A useful result is since the matrices themselves differ only by Lorentz transformations on some of the elements.
In analogy to the bosonic case, we introduce in terms of which we find A generic orthosymplectic transformation can be written one can show that and these are related by the Lorentz transformation (2.43) Some useful identifies are The gravitini are identified in Dirac spinor language using (B.2). Applying this result gives the transformations where / Λ(U ) is the spinorial version of Λ(U ). The transformations for the dilatini (2.30) are a bit more involved.
where we have used the first identity in (2.44) to replaceX with X. A similar expression holds for χᾱ. Converting to Dirac notation gives 6 The β Λ factor is +1 for IIB/IIA * and −1 for IIA/IIB * . The Ramond-Ramond bispinor in Weyl notation is S αβ = −V αM E Mβ . This transforms as One can show that V α N − V αP E P N =Ē N α and translating this to Dirac form gives In the democratic formulation of type II supergravity, we define The above requires the transformation of the dilaton, which is our last field to discuss. Its behavior in super-DFT mirrors its bosonic cousin. It is a superfield Φ that transforms as a scalar density under generalized Lie derivatives (2.52) The generalized superdilaton Φ is related to the supergravity dilaton ϕ by Φ = e −2ϕ sdet E M A . 7 Presuming the superdilaton to transform by a scalar factor Φ = Φ U ∆ , it follows that The factor U ∆ is a priori independent of U M N .

Super non-abelian T-duality
The simplest and most direct situation where we can explicitly see how the OSp transformations of double field theory come about is in the context of T-duality for a supersymmetric σ-model, namely the Green-Schwarz superstring with a non-abelian (or abelian) isometry supergroup G. This situation was fully analyzed by Borsato and Wulff a few years ago [43]. We first summarize their construction and then reinterpret their results in the language of double field theory.

Worldsheet formulation of non-abelian T-duality
Following [43], the starting point is a worldsheet Lagrangian The supercoordinates Z M = (X m , Θ µ ) parametrize a target superspace. The worldsheet metric h ij is presumed to have Lorentzian signature (−, +) and the worldsheet antisymmetric tensor density ε ij obeys ε 01 = +1. The target space tensors G M N (Z) and B M N (Z) are graded symmetric and antisymmetric respectively. 8 Let the σ-model admit a supergroup G of isometries described by supervectors k R obeying [k R , k S ] = f RS T k T . This is a graded commutator, and the isometry label R should be understood to decompose into bosonic and fermionic isometries, R = (r, ρ). We presume that we can adopt a coordinate system where the coordinates Z M factorize into coordinates YṀ on which the isometries act and spectator coordinates Z M , so that k R = k RṀ ∂Ṁ . The superfields G and B decompose as All the dependence on the coordinates YṀ is sequestered in the left-invariant vector fields e R in the usual manner, e R t R = g −1 dg for g(Y ) ∈ G. We review in Appendix C how the above conditions come about. The generators t R obey the algebra Our supergroup conventions are given in Appendix A. When the isometries act freely (that is, without isotropy), the above has a clear geometric interpretation: the coordinates YṀ parametrize the orbits of G on the manifold. When the isometries act with an isotropy group H, then we can (at least locally) take the coordinates YṀ to parametrize the orbits of G/H. 9 The isotropy condition amounts to invariance under g → gh for h ∈ H, meaning that G M N (and similarly for B M N ) must be invariant under the adjoint action of H, It must also project out the Lie algebra h, Non-abelian T-duality is effected by replacing ∂ i YṀ eṀ R with a g-valued worldsheet one-formÃ i R , and adding a term Treating ν R as a Lagrange multiplier, one recovers the original action wherẽ A = g −1 dg is pure gauge. The T-dual model arises if we instead integrate out the one-form A. Working in lightcone coordinates for simplicity, the Lagrangian becomes where we have introduced The addition of the Lagrange multiplier to E RS is the major difference with respect to abelian T-duality. Integrating out the worldsheet one-forms gives the dual model where the new coordinates are Z M = (Z M ,Ỹ R ) with The choice of grading here may seem awkward, but it makes subsequent formulae simpler: where we define E RS as the graded inverse, E RT E TS = δ S R (−) sr . Comparing the expressions for E M N with the formal result (2.41) for a generic OSp(D, D|2s) transformation, we find U can be written as a sequence of three orthosymplectic transfor- The factor U (0) flattens G and B in the isometric directions with the left-invariant vielbein: this occurred in (3.2). The factor U (1) gives the non-abelian factor that replaces E RS with E RS in (3.7c). Finally, U (2) induces the familiar T-duality transformationà la Buscher. Now one can use the results in section 2.3 to compute the new gravitini, dilatini, and Ramond-Ramond bispinors. (We will return to the question of the dilaton in due course.) The additional ingredients we will need are Now we can directly compute the new supervielbein 14) The Lorentz transformation Λ and its inverse are It is difficult to characterize fully this Lorentz transformation, although one can show that det Λ = (−1) dim B G where by dim B we mean the bosonic dimension. This was proven in [43] for a bosonic group. Adapting their proof for a supergroup is straightforward. In their eq. (3.10), promote traces and determinants to supertraces and superdeterminants, leading . Because Ē is the supertranspose of E, their superdeterminants are related as sdet Ē RS = sdet E RS × (−1) dim F G where dim F denotes the fermionic dimension. The result follows since sdet(−1) = (−1) dim G .
The super two-form, covariant field strengths, and dilatini transform as These results match those found by Borsato and Wulff [43] subject to the identifications This argument is perhaps a bit too slick as it appears to ignore a key point: the transformation of E M N does not completely determine U. Put simply, there are as many degrees of freedom in U as there are in V A M itself, but only some of these appear in E M N .
The choice (3.11) was merely the simplest choice that reproduces E M N , but this is hardly conclusive. What actually singles it out (we will show) is that it leaves the generalized fluxes of double field theory invariant -this has the crucial effect that it guarantees the dual theory will possess the proper supergravity constraints.

Double field theory interpretation
In defining the dual coordinateỸ R in (3.9), we have (as usual in bosonic T-duality) identified it with the Lagrange multiplier ν R directly, swapping the index location by hand. This may not actually be the most natural choice; instead, what we can do is to think of ν R as a function of the new coordinates, which we denoteỸṀ . 10 These can be interpreted as the natural DFT coordinates dual to YṀ . Then in the σ-model action, we denote withẽ RṀ is interpreted as the dual analogue of eṀ R . The crucial feature is that while eṀ R is the left-invariant vector field of the group G and therefore carries a flux, the dual vielbeinẽ RṀ is purely flat. This slightly modifies U (1) to For U (2) , we simply replace R withṀ everywhere. Now it will be convenient to denote the indices of these matrices as 10 One could presumably also let νR depend on the spectator coordinates, but this muddies the water.
where M is flattened in the isometry direction, i.e. it involves M , M and R , R . From the perspective of double field theory, we can dispense with U (2) : this merely has the effect of swapping which coordinates we view as physical and which as winding, so we can think of it as a purely passive transformation. What interpretation do we give to U (0) and U (1) ? Suppose we have a generalized vielbein depending on two sets of doubled coordinates, YṀ andỸṀ as well as Z M andZ M , in such a way that it decomposes into a product of two factors: The first factor involves only the Y coordinates and the second only the spectators. (We don't actually need the dualZ M coordinates, but we keep them for generality.) In the bosonic limit s = 0 (3.25) reduced to the generalized Scherk-Schwarz ansatz [74][75][76] in DFT.
Here, we study its natural supersymmetrization. The tilde index M = ( M , M ) decomposes as M = (M , R). We presumeV is chosen so that That is,V is the identity in the non-isometric directions; this is the situation in the case at hand. The original model and its dual differ only in the choice ofV which in the two cases is Here one should think of ν R (Ỹ ) as the potential forẽ RṀ as in (3.22). The first generalized vielbein depends on Y but notỸ , and vice-versa for the second. Both of these, viewed as generalized vielbeins, involve the same flux tensor. Recall that in double field theory, one can build a generalized flux tensor F ABC from the generalized vielbein, with the sign here chosen so that the definition of the flux tensor matches that of the torsion tensor in conventional (undoubled) superspace. Using the decomposition (3.25), one finds where F CBA is built purely from V (which is unchanged under duality) and for both the original and dual models. This suggests an alternative way of seeing that the class of Green-Schwarz superstrings obeying the κ-symmetry constraints (6.1) and (6.2) is closed under super non-abelian Tduality, a result established explicitly in [43]. Let's begin with two observations: • The Green-Schwarz action on its own does not contain all of the physical datait contains only G M N = E M a E N a and B M N . However, if it obeys the κ-symmetry constraints, then one can uniquely identify the gravitini E M 1α and E M 2α , as well as the dilatini and Ramond-Ramond bispinor by imposing various purely conventional constraints on top of κ-symmetry [77]. From these data, one can identify the generalized supervielbein up to its local tangent space symmetries (which include the double Lorentz group).
• The duality transformations from the GS action determine E M N from E M N , but this does not allow one to completely determine the orthosymplectic element U. There is residual ambiguity corresponding precisely to the elements not appearing explicitly in E M N (and thus the GS action) -the gravitini, dilatini, Ramond-Ramond bispinor (plus the extra local gauge symmetries). We merely guessed the simplest form of U.
But these issues are related! The simple choice of U turns out to leave the generalized flux unchanged. Since the κ-symmetry constraints are already encoded in the fluxes, these are maintained as well. Hence, κ-symmetry is preserved under non-abelian T-duality. 11

The role of the dilaton and modified / generalized double field theory
We have not addressed how the dilaton changes under the duality. 12 We will do this momentarily, but first, let us make a brief digression on the subject of what we call generalized double field theory. Recall that the DFT dilaton Φ (here a scalar density of weight 1) can be used to construct a flux Upon solving the section condition, the generalized dilaton is related to a conventional dilaton e −2ϕ via the superspace measure, Φ = e −2ϕ sdet E M A . Just as generalized supergravity [77,78] relaxes the assumption that a dilaton exists, one can define generalized 11 To put it another way, the simple choice of U turns out to be the one that leads to the same choices of gravitini, dilatini, and Ramond-Ramond bispinor made in [77]. 12 From the perspective of the σ-model, the dilaton is an additional field added in order to restore Weyl invariance at the one-loop level. From the perspective of supergravity, the dilaton is a scalar field whose supersymmetry variation gives the dilatini. The perspective here is analogous to the supergravity point of view.
double field theory by relaxing the assumption that a generalized dilaton exists. Then one replaces the flux (3.32) with This is written in terms of a vector field X M which a priori obeys no particular constraints.
In order for F A to be a scalar under generalized diffeomorphisms, X M should transform as What distinguishes the choice of X M is that one requires F A to obey the same properties as it did when the dilaton existed. That is, we impose the same constraints and the same Bianchi identities. Viewed in this way, X M is defined in terms of F A . What exactly does this mean? The flux tensors F ABC and F A obey the Bianchi identities The expression for Z AB (3.35b) can be rewritten in two equivalent ways while Z in (3.35c) can be rewritten as When Z AB and Z vanish, X M has an obvious interpretation as a generalized Killing vector. Generalized double field theory is nearly (perhaps completely) equivalent to modified double field theory (mDFT) [79]. The distinction is that mDFT imposes the section condition on the index of X M , so that X M X M = 0 and X M ⊗ ∂ M = 0. Upon doing so, Z vanishes and Z MN vanishes only if X M is the gradient of some other field. It is unclear to us whether the reverse is true, whether imposing Z = Z MN = 0 necessarily implies the section condition on X M . If it is, then mDFT and generalized DFT should be identical. 13 Both generalized DFT and mDFT lead to generalized supergravity upon solving the section condition, where we define In mDFT, one has both a vector XM and the dilaton gradient ∂MΦ, but in principle one could just absorb the latter into the former to arrive at the formulation discussed here. It was argued in [80] that mDFT can always be interpreted as conventional DFT where the dilaton carries a linear dependence on some of the winding coordinates, while still satisfying the section condition. This forces the generalized vielbein to be independent of those winding coordinates.
The measure factor in the first equation accounts for the inhomogeneous term in (3.34) so that both X M and K M are a conventional one-form and vector respectively. The explicit factor of the B-field ensures that X M is inert under the B-field gauge transformations. The factors of −2 are chosen so that X M = ∂ M ϕ when a dilaton exists. Now one can show that if the modified flux tensor F A obeys the same Bianchi identities and same constraints as before, then the vector K M turns out to be a Killing vector in conventional superspace and X M is a one-form whose spinorial components are the dilatini. The other relations discussed in generalized supergravity [77,78] can be derived in like manner from generalized / modified DFT. We hope to elaborate on this in superspace in the future; the bosonic proof of this was given in [79].
Returning to the original question: how does the dilaton or, more generally, X and K change under duality? Factorizing the supervielbein as in (3.25), the dilaton flux becomes We posit that the dilaton flux should remain unchanged. If so, then the element in parentheses must be fixed. In the spectator directions, we have simply X M = X M and X M = X M .
In the isometry directions, we find more intricate relations where for convenience we have defined Here we will need the explicit transformation of the supervielbein in terms of X M N . This leads to where we have exhibited the dependence on the coordinates Y ,Ỹ , and the spectator coordinates Z. From these relations, we find Now we may identify the X and K fields. In the original model, we take In the dual model, we have the somewhat more complicated expressions Here we must remember that the duality involves a passive coordinate transformation so Rather then perform index gymnastics with the isometry coordinates, we will simply express relations in terms of the flattened isometry index, even though it is in the "wrong" position: A rather strict check of these relations is this: K X should vanish in the dual model when it vanishes in the original model. This is a consequence of T-duality preserving κ-symmetric Green-Schwarz actions. We find The second line can be rewritten as The first term drops out immediately using the Jacobi identity. To evaluate the remaining terms requires a few features of K and X that arise in generalized supergravity. First, G M N and B M N , once flattened as in (3.2), are independent of YṀ . The same should be true of K and X in generalized supergravity, because their various components appear in the torsion and curvatures. 14 This means that the isometry condition on G M N implies in particular that (3.52) 14 We are speaking here of the flattened versions K R and XR.
For the B-field, the relevant relation we need is dX = −K H. From this, one can show that Taking the difference between these two relations lets one rewrite the right-hand side in terms of E = G − B. Introducing the Lagrange multiplier field converts E RS to E RS . Using the Jacobi identity simplifies the result to where we have suppressed gradings in the final term for readability. The term on the left-hand side is exactly what remains in (3.51). Substituting this expression, we find the complete cancellation of the remainder of the right-hand side of (3.50).
A specific case of interest is when we start with a model with a dilaton, a case analyzed in [43]. Then K = 0 and X = X = dϕ. The equations (3.49) can be rewritten where we have used D R ϕ = 0. Now the dual theory satisfies the conventional supergravity constraints when This imposes a requirement for how the dilaton should depend on the coordinates we are dualizing. To solve this, we could extract from the dilaton a purely Y -dependent piece that generates this term, i.e.
In general there is no local obstruction to the existence of ∆, since it obeys the consistency condition [D R , D S ]∆ = −f RS T D T ∆ by virtue of the Jacobi identity. Now the dual dilaton can be identified as so that X = X = dϕ .

Component description
The previous discussion has been at the level of superspace. In order to make contact with the literature on fermionic and bosonic T-dualities of bosonic backgrounds, we should rewrite our expressions at the component level. Here we must already make a distinction between bosonic and fermionic isometries that arise from the algebra of supervectors • Bosonic isometries are treated as conventional vectors k r = k r m ∂ m acting on bosonic coordinates. These arise by taking the θ = 0 parts of the bosonic supervectors k r = k r M ∂ M . Since k r µ is fermionic, it must be at least linear in θ, and so can be discarded.
• Fermionic isometries are described by commuting spinors ε ρ iα with i = 1, 2. These arise by flattening the fermionic isometries k ρ with the gravitino one-forms and setting θ = 0: Since k ρ m is fermionic (being linear in θ), it can be discarded.
As is well known, bosonic isometries can arise as bilinears of fermionic ones. To describe this, we first rewrite (3.58) with flat indices. Under a covariant Lie derivative generated by k R , the supervielbein is merely rotated, where λ R is a Lorentz transformation. This follows from the Green-Schwarz action, since invariance of G M N implies the result for E M a ; for E M iα , one must employ the torsion constraints (which arise from κ-symmetry). This expression may equivalently be written The algebra of Killing supervectors can then be rewritten (with gradings suppressed) These expressions lead immediately to several useful results. First, taking (3.62) with A = a and RS = ρσ, we find how a bosonic Killing vector is generated from two Killing spinors: where k t a = k t m e m a . The chirality of ε 1 is fixed while that of ε 2 depends on whether one lies in a IIB/IIB * or IIA/IIA * duality frame. A crucial point is that the fermionic indices appear symmetrically in (3.63) and the two commuting spinors may be taken to be the same: Taking A to be spinorial in (3.62), we find the other useful relations with / F given by (2.50).
The vielbein and B-field expressions match what the computations purely from the bosonic dualities would give, indicating that the fermionic T-dualities have no effect on them [48]. Where the fermionic T-dualities matter is for the Ramond-Ramond background and for the dilaton, where we find The additional Lorentz transformation above is given in vectorial form as and depends purely on the bosonic isometries.
The expression for the Ramond-Ramond bispinor involves E ρ 1α = e ρ E 1α , but it would be more useful to rewrite this in terms of k ρ E 1α = ε 1α ρ . To do that, we need to apply the adjoint action of g to the isometry indices. Recall we have In choosing the original coordinate system (3.2), we expanded in terms of the left-invariant vector fields. The right-invariant vector fields are dgg −1 = dYṀ kṀ R t R and these are related to eṀ R by Applying the adjoint action to E RS gives where we suppressed gradings in the first equality. Since we only care about purely bosonic expressions, we have simply The dilaton can be written The Ramond-Ramond bispinor becomes An extra sign has appeared because we use the inverse (Q −1 ) ρσ rather than the graded inverse.
What can we say about Q ρσ ? While it is fully characterized in superspace, on the bosonic background it can really only be described by its derivative. From the definition of Λ ρσ in superspace, one can show that This follows because the B-field in (3.2) (like the metric) obeys L k R B = 0. The more general case is discussed in Appendix C. This leads to The quantity Q ρσ should depend on the spectator coordinates, the dualỹ coordinate (via ν u ), and on the coordinates y only via the adjoint action (since E ρσ was y-independent). This means The terms involving ν already have this form, since k r dν u = 0 and the the Jacobi identity allows one to rewrite the pair of structure constants appropriately. For the terms involving H and B, it helps to observe that k R H = −d(k R B) from which the desired property can be deduced. A key step is to exploit which follows from the explicit form of H in terms of the B given in (3.2). The expression (3.79) can be interpreted purely as a bosonic equation once we address the first term involving H. It is given by Note the crucial relative sign difference with (3.63). The importance of this sign difference was already noted in the context of non-abelian fermionic T-duality in [53].
Abelian fermionic T-duality. The fermionic T-duality discussed by Berkovits and Maldacena [48] corresponds to a single abelian fermionic isometry, for which the left-hand side of (3.63) vanishes. No bosonic isometries are involved and so the vielbein and B-field are unchanged. However, the dilaton and Ramond-Ramond complex change as Since there is no duality in a bosonic direction, there are no dual bosonic coordinates for Q to depend on.
Non-abelian fermionic T-duality. Slightly more generally, one can consider a single non-abelian fermionic isometry [53], which generates a single bosonic isometry: Because we must dualize the full set of closed Killing supervectors, this is actually two dualities: a fermionic one k ρ and a bosonic one k r . In our conventions, This can perhaps more transparently be written in the following way: whereẽ rṁ is the dual vielbein. Note that k r V = 0. This is apparent both from the explicit expressions in terms of the Killing spinors but also from (3.81) which collapses to k r k ρ k ρ H = 0. The expression for ∂ m Q ρρ in (3.89) matches the result in [53] (with Q ρρ → C and B → −B) but the expression for ∂ m Q ρρ is different, with k r m there in place of ourẽ rṁ . (In our approach, ∂ m Q ρρ vanishes since there are no dual coordinatesx m in the σ-model.) This is actually a possibility, because in the case of a single bosonic isometry, one can choose coordinates in the original geometry so that k rṁ is a constant. Then one simply takes ν r = k rṁxṁ . Next we address the vielbein and B-field. They are where we have exploited that (Ad g) r r = 1. Finally, the Ramond-Ramond complex is Recalling that G rr = k a r k ra , the Lorentz transformation governing the T-duality frame is Since this is a single bosonic T-duality, it exchanges the type of supergravity, from type IIB/IIB * to IIA/IIA * . One finds det Λ = −1 using the general argument reviewed in section 3.1. Whether this exchanges the star type (e.g. from IIB to IIA * ) depends on whether Λ 0 0 is positive or negative, that is whether the T-duality is spacelike or timelike.
We find which is indeed positive for spacelike k r and negative for timelike k r .

Construction of V A M for constant fluxes
In the preceding section, we focused on σ-models depending on two sets of fields: spectator fields Z M and fields YṀ that were freely acted upon by some set of isometries. After performing non-abelian T-duality, we arrived at a model with dual fieldsỸṀ . A key point we emphasized was that the dualized sector admitted a double field theory interpretation, with two different generalized vielbeinsV, (3.27) and (3.28), depending respectively on Y andỸ , so that the generalized fluxesF (3.31) were identical and constant. Let us focus on this last point first, and for simplicity, we dispense with spectator fields. In analogy with the bosonic analogue [20,28,59], we define a generalized parallelizable superspace as a (D + s)-dimensional super manifold upon which we can introduce a set of OSp(D, D|2s)-valued generalized frame fields V A M whose generalized flux tensor F ABC We will not discuss the second question here, but such a model does exist: it is known as the E-model [13] and corresponds essentially to the Tseytlin duality symmetric string [81,82] with the generalized metric V A M given below. We refer the reader to the original literature as well as the recent discussion in [33].
To construct the requisite V A M , it turns out that just three conditions are sufficient: Different choices of H turn out to correspond to different dual geometries and the supervielbein describes a coset H\D.
For the case of non-abelian T-duality discussed in the previous section, we would have The t R generate the isometry group G andt R generate an abelian dual groupG. The original σ-model geometry is produced by choosing H =G and the dual geometry is produced by H = G. This case is known as a Drinfeld double, since the quotient of D by either maximally isotropic subgroup G orG generates the other group, i.e. G =G\D and G = G\D. The duality exchanges the roles of G andG. It is also possible for both groups G andG in a Drinfeld double to be non-abelian. This leads to Poisson-Lie T-duality [17], and this was historically the first step in generalizing non-abelian T-duality. The construction of V A M proceeds as follows. A general group element of D is denoted g, and its left coset M = H\D corresponds to a decomposition g = h(ỹ) × m(y). The generalized frame field is built from m. First, we decompose the Maurer-Cartan form as where V A and A A are valued respectively on the coset and the subgroup. Next, we build the two-form B WZW by integrating This is usually only locally defined. Then the generalized frame field is given by We have denoted the two-form by B rather than B since contributions from M A B typically deform the matrix structure and contribute to the physical B-field.
For the case of non-abelian T-duality, choosing H = G leads to where e R M are the left-invariant vector fields on G, see (3.27). Alternatively, one can Swapping indices around, one can show this is just where U (2) and U (1) are the subblocks of (3.24) in the isometry directions.
More interesting examples are possible for any real Lie supergroup G, provided it admits a non-degenerate Killing form. These can be extended in two distinct ways to a double Lie group D, either by taking the product group G × G or its complexification G C .
Both of these cases will be extremely important for the remainder of the paper, and we will describe them in some detail.

Example
We denote a group element of D = G × G with the tuple (g 1 , g 2 ) ∈ D with g 1 , g 2 ∈ G. We use the same convention for the Lie algebra d to define the pairing for Ξ = (ξ 1 , ξ 2 ) ∈ d. In terms of the generators t A of G, we choose the basis of generators on the product group as In the second set, we have raised the indices using the graded inverse κ AB (with NW-SE conventions) of the non-degenerate Killing form κ AB = t A , t B . This choice guarantees that T A , T B = η AB and that T A generates the maximally isotropic subgroup H = G diag . This is in fact the only viable choice without imposing additional structure on G. The resulting coset M = H\D is isomorphic to G. The structure constants F ABC , defined by are given by A convenient coset representative is M m = (g, e) , g ∈ G , (4.14) where e is the identity element in G. With this convention, it is straightforward to compute all the ingredients for (4.5), namely Above we employ the right-invariant vector field v A on G. To write down the resulting generalised frame field in a simple form, we also introduce the left-invariant vector fields on G, and the respective inverses v A M and e A M with the defining properties 20) to eventually obtain which is related to (4.14) by an H transformation, m = hm = (g, e) = (hg , hg −1 ) for g = g 2 . Explicitly, we find and the generalised frame field arises by plugging these quantities into (4.5). The resulting frame is related by a diffeomorphism (and B-field gauge transformation) induced by g = g 2 to the frame in (4.21), using This can equivalently be written Explicitly, we find the expression where now we interpret v M A and e M A as the one-forms that solve (4.26). Naturally this is the same expression as (4.21), merely interpreted differently, in a different coordinate system. Note that we still have Though rewriting it in this way may seem to needlessly complicate matters, it will actually make it easy to see how the generalised frame on G C , which we construct next, can be related to G×G by analytic continuation. The key feature of the coset representative consistent with the relations between T A in the two cases, provided we transform ∂ M → (− ∂ M , ∂ M ). That is, we flip the sign of x but not of the dual coordinate. This is sensible, since the dual coordinate parametrizes the diagonal subgroup, which is quotiented out by the coset and undergoes no change. with X, Y ∈ g C . In this case a natural choice for the pairing is where X and Y are elements of the complexified Lie algebra g C . Here, we in particular make use of the Cartan involution σ with the properties It specifies how the real Lie algebra g is embedded into g C by identifying the former's generators t A with the +1 eigenspace of σ, i.e. σt A = t A . We further assume that σ is given by σX = −S −1 X † S where S denotes an optional similarity transformation (for compact G, we can set S = 1). This implies that the structure coefficients are (graded) real, meaning (f AB C ) * = f AB C (−) ab . The same holds for the Killing metric (κ AB ) * = κ AB (−) ab .
For the generators of D, we are going to explore two distinct cases. The first is obvious: with non-vanishing components of the generalised flux F ABC given by For the coset representative, we take a Hermitian element of G C , so that σm = m −1 . Effectively, we can think of m = exp(ix A t A ). The building blocks of the generalized vielbein are then We introduce the one-form e A and its complex conjugateē A , The primes are for later convenience as these will be related to e and v in the previous section. For these we recover Now the full generalized vielbein can be written where we use 42) or equivalently, This case and the one for G × G with coset representative (4.22) are related by an analytic continuation. There are several ways of seeing it. From the level of the building blocks (4.36) -(4.38) and the algebra, we can see it by continuing T A → iT A . To maintain η AB , we must substitute ·, · → −i ·, · , too. Consequentially, we obtain while for the two remaining constituents of the generalised frame field we find This is somewhat formal, and we can make it more concrete by observing that both coset representatives m are inverted by their respective involutions, and we use this involution to track how factors of i are inserted. Here, m = exp(ix A t A ) and for (4.22) we have g = exp(x A t A ). We want to analytically continue by taking x = ix. By comparing explicit formulae, we see that D (x ) = D(x) and so e M A (x ) and v M A (x ) become, respec- Putting this together we see that the two generalized vielbeins V A M turn out to be related by consistent with the relations between T A in the two cases, provided we identify ∂ M = (−i ∂ M , ∂ M ). That is, on the doubled space, we transform x = ix but leave the dual coordinate unchanged. This makes sense on the coset since the dual coordinate describes a copy of G itself in both cases (being the same isotropic subgroup H), and undergoes no analytic continuation. There is another possibility that will be of interest to us, 16 for a matrix R AB obeying certain properties. Requiring T A , T B = η AB implies that R AB is graded real and antisymmetric. Requiring that T A generate a maximally isotropic subgroup implies where we employ operator notation for R, i.e. R · ξ = ξ A R A B t B . From this equation, we learn that R must solve the modified classical Yang-Baxter equation (mCYBE). For the 15 The forms pick up factors of i because dx = idx. 16 The decomposition (4.48) is actually a Drinfeld double, and one could exchange the roles of TA and T A . The result is essentially equivalent to taking (4.34), up to a similarity transformation and coordinate transformation.
coset representative m = g, which is now fixed by the involution σ, we again compute all ingredients required for the generalised frame field, We can streamline the result further by defining 52) and the corresponding dual vector fields v A M and e A M (see (4.20)). With them, we eventually find It is useful to make a similarity transformation on the generalized vielbein and the generators in this case to give and with generalized fluxes Up to the interchange of T A T A , the generalized vielbein (4.57) and the one constructed from (4.36)-(4.38) are Poisson-Lie T-dual to each other.

The role of the dilaton
We have not yet discussed the role of the dilaton on a generalized parallelizable space. Let us address this briefly now. In terms of the generalized dilaton Φ, the dilaton flux is given by (3.32), which for the supervielbein (4.5) becomes In the case of generalized double field theory, we replace ∂ M log Φ → X M and relax the section condition on the free index of X M . Solving for X M = (X M , X M ), we find The dilaton is not completely arbitrary since we still require F A to obey the usual Bianchi identities. In the context of a generalized parallelizable space, when the fluxes F ABC are taken to be constants F ABC , the most natural choice is to take the dilatonic fluxes to be constants as well, F A = F A . The Bianchi identities then imply F AB C F C = 0, and the conditions (4.60) simplify to These can be interpreted as solutions for the vector X M . In order to admit a dilaton solution consistent with the section condition, one must restrict F A = −F AB B . As a special case, we can consider both G × G and G C . For G × G using the coset representative (4.14), we find with F A and F A obeying A dilaton solution requires F A = 0. For G C using the coset representative g in the basis (4.48), we find with F A and F A obeying If we make the similarity transformation to the simpler basis (4.56) with T A = i κ AB t B instead, one replaces F A with F A + R AB F B in the above formulae. To admit a dilaton solution, we must have the following condition (4.66) 5 Generalized supercosets

Review of conventional supercosets
To motivate the construction of generalized supercosets, we first recall how conventional supercosets are constructed. Let G be a group and F be a subgroup. Denote the generators of G by t A , the generators of F by t A , and the remaining generators by t A . The structure We decompose a generic group element g as g = m(z)f (y) with coset representative m. The local coordinates are chosen as z M = (z M , y I ). The Maurer-Cartan form dz M E M A t A = g −1 dg decomposes as This decomposition shows how the full group can be reconstructed from the coset. In particular, it has three important properties: 1.) All quantities relevant for the coset are contained in the middle matrix in (5.1). These depend only on the physical coordinates on the coset.

2.) This matrix is in upper triangular form.
3.) It is dressed by an adjoint f ∈ F action on the right and right-invariant Maurer-Cartan form of the subgroup F on the left. These depend only on the subgroup coordinates.
With the dual vector fields corresponding to (5.1), one can compute the anholonomy coefficients With the y coordinate dependence isolated in the first and third factors, one can show that the anholonomy coefficients with a lower index valued in F are completely fixed in terms of the structure constants. Up to an adjoint action of f , which we discard in the definition of F A B C , we find while the remaining two correspond to the covariant torsion and curvature tensors The results (5.5) and the covariance of (5.6) follow from the general fiber bundle structure of (5.1) with local symmetry group F acting on the frame bundle. When E M A and Ω M A are determined from a larger group G, the covariant torsion and curvature tensors are fixed as

Generalized supercoset construction
Let's apply similar considerations to the case of a double Lie group D. As before, we presume a maximally isotropic subgroup H, consistent with the assumptions made in section 4.1. We denote the generators of D as T A = (T A , T A ) with T A the generators of H.
In addition, we presume that D possesses another isotropic subgroup F , with generators T A , with respect to which we will construct a generalized coset H\D/F . There is a subtlety here, which we should address at this point. We make no assumptions about how F and H are related. This means we will need two distinct bases for the generators of D: the original basis T A = (T A , T A ) where T A are the generators of H, and a new basis, where T A are the generators of F . For this new basis, we take the Killing metric to be with η AB an OSp metric on the coset. The change of basis matrix between T A and T A may in principle be quite complicated. To avoid a proliferation of indices, we won't explicitly exhibit the prime on A, but it should be understood to be in the appropriate basis. On the generalized frame field in (4.5), we aim to impose a similar decomposition inspired by (5.3). The role of the group G and subgroup H will be played by the left coset H\D and the subgroup F respectively: By preserving the OSp pairing on the generalised tangent space and splitting it into coset and subgroup contributions, we obtain and Ω BN = Ω N B (−) nb and the graded antisymmetric matrix ρ AB . The former plays obviously a similar role as the connection Ω M A in the geometric coset, while the latter is a new ingredient required only in generalised geometry. Remarkably, ρ AB also appears in the work by Poláček and Siegel to construct a natural curvature with manifest Tduality [83]. There, the subgroup F is the double Lorentz group and the contracted version ρ EF F EAB F F CD = r ABCD is used. Hence, we call ρ AB the Poláček-Siegel (PS) field.
From now on, we will refer to V A M as the megavielbein and the enlarged superspace on which it acts the megaspace, when we need to distinguish it from the coset supervielbein V A M . Similarly, we use to denote their respective flat derivatives. From (5.10) and recalling that ∂ I vanishes, the flat derivative on the megaspace becomes Just as in the conventional supercoset, the middle matrix in (5.10) depends only on the coset coordinates. With the y coordinate dependence isolated in the first and third factors, one can show that up to an overall adjoint action of f , which we discard, the generalized fluxes with a lower index valued in F are completely fixed as 14) The remaining fluxes correspond to generalized curvature tensors. The torsion tensor is where F is the generalized flux of the coset supervielbein. The generalized Ω curvature is Finally, there are two additional curvatures that are not present in the standard supercoset. These are the covariantized gradient of the Poláček-Siegel field and an additional curvature where we have shown only the leading terms. The precise forms of these curvatures will not be relevant for us. The curious reader can find them in [38,42]. These generalized torsion and curvature tensors hold for generic V, Ω, and ρ. When they are determined from a larger doubled Lie group, they are constrained as Now let us compute the quantities (4.5)-(4.7) required to construct the generalised frame field for the coset representative m = nf , where f is an element of an isotropic subgroup F generated by T A . We find the adjoint action the Maurer-Cartan form components and the B-field Above we have split the original B WZW from (4.7) into an exact term and a term B WZW defined purely on the coset. A straightforward calculation gives rise to where v A = v A I ∂ I denotes the vector field dual to the one-form dy which proves that holds. This verifies the form of the last column in the middle matrix of (5.10). But because V A M is an OSp element, the first row also has the desired form. We can finally read off where we introduced for convenience the quantity It is a somewhat involved calculation to show that both S A B and V A M are y-independent, while B N M and V M A are y-independent by construction.

The dilaton on the generalized supercoset
Now we will equip the Poláček-Siegel megaspace with a dilaton Φ. Its generalized flux tensor is In analogy to the decomposition of the megavielbein (5.10), we expand log Φ = log Φ + logẽ (5.32) whereẽ A T A = f −1 df is the left-invariant vector field on F and Φ is chosen to be independent of y. The extracted term is responsible for generating the density behavior of Φ under y diffeomorphisms. One can now show that are the dilatonic torsion and curvature respectively. In the case of generalized DFT, one should replace ∂ M log Φ → X M in (5.31). A natural replacement of the constraint (5.32) is where X M and X A transform under coset diffeomorphisms and F -gauge transformations as Now the dilatonic torsion and curvature are The dilaton solution corresponds to X M = ∂ M log Φ and X A = 0 where Φ is gauge invariant under F .

Example: D = G × G
The examples we will consider are based on the ones presented in the previous section, namely G × G and G C . We employ the same real semisimple Lie group G as before, but additionally, we presume the existence of a subgroup F ⊂ G. The most relevant cases are when the coset G/F is a symmetric space, but we will remain rather general here.
When embedded into the double Lie group D = G × G, the subgroup F must be isotropic. The pairing (4.10) makes this constraint very restrictive and only allows for diagonal subgroups. We denote the generators T A = (t A , t A ) for the generators of F . In other words, F here is a subgroup of H itself. The remaining generators are assigned by requiring that the pairing η A B has to be of the form given in (5.9) and we get There is a subtle point here: in defining the left coset, H\D, we arranged the generators as , with the latter defining H. In defining the right coset now, we have swapped the roles of lower and upper indices. 17 Now we can build the components of the generalized vielbein. As the coset representative, we take with f ∈ F and n in the dressing coset G diag \(G × G)/F . Because F ⊂ H, some care must be taken in the choice of n, because this coset representative may be rewritten as The factor on the left is an element of H, so its only effect is to add an exact term to the B-field. For this to be a good coset representative, we must be careful to choose n so that f −1 nf is a sufficiently generic element of G -namely, that it generates invertible left-invariant and right-invariant vielbeins. This is not always possible -e.g. if F contains an abelian factor that commutes with all elements of G. 18 In fact, the coset representative (5.42) is nothing but the coset representative used in (4.14) for the case g = f −1 nf . This means that the generalized vielbein we will construct must actually be equivalent to the generalized vielbein there (4.21), up to an exact shift in the B-field, and an overall OSp transformation acting on the left to swap the roles and index positions of T A and T A .
We can begin to see this already when we compute V M A : 17 We additionally could swap the roles of TA and T A (raising/lowering the indices respectively) to restore the original positioning of the coset indices, but this only works if κAB vanishes, since we need TA, TB = 0. 18 It is even problematic for symmetric spaces if we choose n = exp(x A tA), since then the effect of f is merely to rotate the coordinates x A . Then the left and right-invariant vielbeins vanish on the subgroup F since there is no dy component.
It is nothing but the adjoint action of f on the right invariant vector field of g. More explicitly, we take where S A B was defined in (5.30). Importantly, we will need the two conditions We need to compute M A B . Here one needs to keep in mind that the A index is in the F -adapted basis, whereas the B index is in the H-adapted basis. This leads to The vector pieces of the generalized vielbein are At this point, we denote the coset part of the inverse Killing metric κ AB , which we presume to be invertible with graded inverse η AB , Note that η AB does not equal κ AB unless κ AB vanishes. Now on the coset, we introduce the vector fields We presume these are invertible. Then we find At this point, we can exploit a fact more familiar from O(D, D) elements that can be extended to OSp elements when we have a metric κ AB with inverse η AB . In general, we may write  (5.56) or equivalently, For reference we give the translation between dnn −1 and n −1 dn and the 1-forms e A and v A introduced on the coset: The two vielbeins are related by a graded version of a Lorentz transformation, where explicitly The remainder of the megavielbein is characterized by Ω and ρ: It will actually be useful for us to consider a slightly different coset representative, which will be relevant for analytic continuation: The coset element (n , n −1 ) goes to its inverse under the involution σ that exchanges left and right group factors. Thankfully, we do not however need to perform any new computation. Similar to the generalized group manifold case, this coset representative is related to the previous one merely by an H-action on the left (which is just an exact shift in the B-field) and a coordinate transformation, taking n = n 2 , exploiting the identification Of course, it is related to the two G × G generalized group manifold cases as well.
With these facts in mind, and using what we have learned in the previous cases, we can simply describe the result here in a manner that will be useful for analytic continuation.
Let g = f −1 nf be a generic element of G, and similarly for g = f −1 n f with n = n 2 (so g = g 2 ) . Define on the full group the modified left and right invariant forms In terms of n , these can be written These equations imply that Then the generalized supervielbein on the large space is given by (5.10). The connection Ω and Poláček-Siegel field are 5.76) and B is given by 5.5 Example: D = G C Next, we take the complexified group D = G C discussed in section 4.3. The subgroup F ⊂ G is again an isotropic subgroup using the pairing (4.32). The basis (4.48) already introduced for the G C case is perfectly suitable here: we merely split the generators up so that Again, we do not need to impose that κ AB vanishes, although this will certainly be the case most of interest. A natural coset representative lies in G itself, Introducing the usual left invariant vector fields suitable for G/F , This leads to a generalized vielbein on the coset of The connection Ω and Poláček-Siegel field are The matrices Π A B appearing above are given by Note that Π A B resembles the matrix Π A B given in (4.53), except we have restricted the group element used to construct D A B from g to n. Of course, this is no accident: the megavielbein on the generalized coset is nothing but the generalized vielbein on the full space (up to a B-field gauge transformation). It is an instructive exercise to check these formulae emerge directly by comparing with the expression (4.53) and extracting (Ad f ) A B .
Just as on the generalized parallelizable space, we can make a similarity transformation to the basis This remains in Poláček-Siegel form, except now the various constituents of the megavielbein are given by , This case can also be easily compared with the corresponding megavielbein in (4.57) after extracting a factor of (Ad f ) A B .
Another interesting case is to choose We use the same decomposition for m as given in (5.80) with n being hermitian and f unitary. For the elements of M A B we find The computation is very nearly identical to the G × G coset. We find for V A and A A

91)
The vector pieces of the generalized vielbein are and similarly for its complex conjugate, Inspired by G × G case, these can also be written The generalized supervielbein on the coset is The Ω connection and Poláček-Siegel field are 6 Generalized supercosets for supergravity backgrounds

Supergravity backgrounds in double field theory
In order for the generalized supervielbein to describe a valid background of supersymmetric DFT, the generalized flux tensor must obey a certain set of constraints [38] (for earlier work, see [39,40] and [41]). At dimension -1/2, all flux tensors vanish while at dimension 0, We refer to these as κ-symmetric constraints, in analogy to their supergravity analogues [77]. In addition, one imposes conventional constraints at dimension 1/2 which amount only to redefinitions of the physical dilatini and gravitini. A final conventional constraint at dimension 1 redefines the Ramond-Ramond bispinor, As argued in [38] (and in analogy with [77]), these constraints alone lead to a generalized double field theory (which is related to modified DFT [79]), the DFT analogue of generalized type II supergravity, where one does not presume a dilaton to exist, see section 3.3. We will return to the question of conventional supergravity (i.e. where a dilaton exists) in section 6.5. Now we can pose the question whether the generalized vielbeins we have constructed in previous sections, namely for the double Lie groups G C and G × G, satisfy these constraints so that they describe supergravity backgrounds. If we presume that the group G should have 32 supercharges (to accommodate the full range of α andᾱ indices we seek), ten corresponding translation generators P a , and a subgroup F corresponding to any Lorentz and/or R-symmetry groups, we are essentially restricting our attention to maximally supersymmetric type II backgrounds. These were analyzed long ago [84], with only the AdS 5 × S 5 background of IIB and its Penrose limit (an Hpp wave) [85] relevant to us here. 19 The supergroup G of isometries for AdS 5 ×S 5 is PSU(2, 2|4) (see e.g. [88]). Only some of the details of this algebra are important to us, so we will treat it in rather general language. It consists of generators t A = {t a , t α , tᾱ, t r }. The generators t r span a (bosonic) subgroup F = SO(4, 1) × SO (5). The generators t A = {t a , t α , tᾱ} comprise spatial translations and supersymmetries, and the supercoset G/F is a superspace whose bosonic body is AdS 5 ×S 5 . The superalgebra admits a Z 4 grading under which t r , t α , t a , and tᾱ carry charge 0, 1, 2, and 3. The non-vanishing (anti)commutators are We normalize the generators so that the SUSY algebra is conventional with Then the structure constants f AB C may be interpreted as the torsion tensor T AB C of the undeformed AdS 5 × S 5 background. The algebra admits a non-degenerate Cartan metric κ A B with nonzero pieces κ ab = η ab , κ αβ = −κβ α , and κ rs . The (graded) inverse component κ αβ is proportional to the Ramond-Ramond bispinor of the undeformed AdS 5 × S 5 background, i.e. κ αβ ∝ F a 1 a 2 a 3 a 4 a 5 (γ a 1 a 2 a 3 a 4 a 5 ) αβ , since it appears in the constant torsion A crucial feature of κ αβ is that due to the 10D gamma matrix identity γ a γ b 1 b 2 b 3 b 4 b 5 γ a = 0, one finds T aβγ (γ a )γβ = T aβ γ (γ a ) γβ = 0.

The η-deformation
In the context of supercoset sigma models, the η deformation is a specific deformation that preserves the classical integrability of the original model. It depends on the existence of an R-matrix obeying the modified classical Yang-Baxter equation (4.49); such models are known as (inhomogeneous) Yang-Baxter σ-models [65,66]. For the case of the AdS 5 × S 5 superstring, the Lagrangian is given by [63,64] The group element g is an element of PSU(2, 2|4). The factor 1/t can be interpreted as the string tension T . The Lie algebra operator d is defined in terms of Z 4 graded projectors as d = P (1) + 2 1−η 2 P (2) − P (3) . As a diagonal matrix, d A B and its transverse are given by 20 The operator O − and a related operator O + are given in matrix form by The Lagrangian (6.8) can be rewritten in Green-Schwarz form as Ref. [68] relates operators to matrices as This amounts to replacing It is straightforward to show that if one decomposes g = nf for f ∈ SO(1, 4) × SO(5), the f factor drops out, so this is indeed describing the supercoset.
In the seminal work [68], Borsato and Wulff analyzed the supergeometry of the ηmodel, establishing that its κ-symmetry was of the GS form and deriving a condition on the R-matrix (dubbed a unimodularity condition) for the background to be a supergravity solution. Our goal in this section is to analyze the η-deformed model purely on group theoretic grounds and show how the relevant structures of the σ-model emerge purely from the doubled supergeometry.
The starting point is the complexification G C of the group G = PSU(2, 2|4). As we have already discussed in section 4.3, the complexified group involves the addition of generators with Killing form built from imaginary part of the Killing form on G, so that (6.14) We want to find a new basis for this supergroup, for which the structure constants can be interpreted as generalized flux tensors for a supergravity background. Denote the generators of this new basis T A = (T r , T A , T r ) with pairing The generators T A = (T α , Tᾱ, T a , T a , T α , Tᾱ) will parametrize the generalized supercoset with pairing η AB given by (2.22). A few basic assumptions will help us choose these generators: • The only group invariant is presumed to be the Killing superform. This suggests that the new basis of generators T A should be very simply written in terms of the old basis, where a, b, c, and d correspond to numerical constants and no summation on the parenthetical indices is assumed. This implies that the flux tensors will all be proportional to the original structure constants, • T r = t r , in order to preserve the coset interpretation, with the Lorentz generator acting on all other generators in the expected way.
• The structure constants must obey the supergravity constraints. This means that all the dimension -1/2 components vanish, F αβγ = F αβγ = F αβγ = Fᾱβγ = 0. This is automatic because there is no corresponding structure constant in the original algebra (since the structure constants are bosonic quantities). The dimension 0 components should also be constrained to obey Additional constraints apply at dimension 1/2; however, these are fermionic and must vanish since the fluxes correspond to structure constants of a supergroup (just as for dimension -1/2). Finally, at dimension 1, we will also require (6.4).
The most general possibility for T α and Tᾱ is We choose an arbitrary parameter η and normalization a 1 to define T α . The fact that −η appears in Tᾱ is a direct consequence of T α , Tβ = 0. From the basic dimension zero flux constraint (6.17), we can deduce T a from {T α , T β } and similarly for T a : The dimension zero flux also fixes T α using [T α , T b ] (and similarly for Tᾱ) as The Lorentz generator and its dual can only be T r = t r , T r =t r (6.18d) in order to satisfy T r , T s = δ r s and T r , T s = 0. From T a , T b = η ab = η ab and T a , T b = η ab = −η ab , we find the normalizations . (6.19) This fixes the range of η as 0 < η < 1 or η < −1. We fix the phases of a 1 and a 2 by choosing them to be positive real numbers. We summarize the full set of structure constants in Appendix D.
There are two equivalent paths to the supervielbein, depending on whether we want to view it as the supervielbein for the generalized parallelizable space (section 4.3) or for the generalized coset (section 5.5). While the most direct path is the latter, it will be more instructive to use the former construction to generate the megavielbein directly, since this is closer in spirit to the results of [68]. Recall that for G C , we gave a simple form for the generalized supervielbein in the basis t A andt A = i t A in (4.57) (promoting unhatted indices to hatted ones). The construction involved the left-invariant vector fields e A t A = g −1 dg and the R-matrix R A B obeying the mCYBE (4.49). Then one simply can apply the dictionary derived above for relating t A andt A to the generators T A we actually want. This gives a simple similarity transformation which can be applied to give the generalized supervielbein.
Actually, in order to match normalizations, we need to rescale the generalized supervielbein with a dimensionful parameter (this is related to rescaling the worldsheet tension): The W factor rescales the flat indices, with nonzero entries The parameter v carries mass dimension, and the choices above reflect the engineering dimensions of D A . The U factor rescales the dual derivative ∂ M , (6.22) The choice of v −2 here is needed to ensure that V remains an OSp element with unchanged η A B and η M N . We drop the prime from now on. After this redefinition, the fluxes are unchanged except for an overall rescaling by v consistent with their engineering dimension.
To match conventions in [68], we will choose v = 2η 1 − η 2 . (6.23) The generalized supervielbein can be read off the covariant derivatives. Using the matrices (O ± ) A B introduced earlier they are D r = e r M ∂ M , (6.24a) It is worth emphasizing here that (O + ) α B = (O − ) α B and similarly forᾱ; this is apparent from the operators themselves, but it is a requirement from the underlying structure of supersymmetric DFT, see the second line of (2.33). The supervielbein implicit in (6.24) is not immediately written in Poláček-Siegel form. In particular, it has dependence on the subgroup coordinates y. However, it is easy enough to put it into that form. Decomposing the group element as g = n × f , the G vielbeins e A M employed above can be rewritten as with e and ω defined as in (5.81). Conjugation by Ad f leaves the diagonal matrices d and d T invariant and replaces R g with R n . This leaves an overall Ad f on the very outside of the megavielbein as in (5.10). The fields on the coset simply correspond to replacing g with n in the operators O ± and dropping the Ad f factor in (6.25). We denote O ± as the operators (6.10) with g replaced by n. The result coincides with applying the similarity transformation for T A to the coset supervielbein (5.88) directly. As discussed in section 2.3, one can read off from these the components of the physical supervielbein. First, one identifies 21

26a)
The fact that it is e B M rather than e B M appearing here is a consequence of the triangular form of (6.25). Their inverses are It is crucial that (O −1 ± ) s A = 0 for the inverses to have such a simple structure.
The OSp structure requires that E M a and E M a be related by a Lorentz transformation, The fact that the index sum is over B and not B comes from the upper triangular structure of e A M in (6.25). One could equivalently write That this matrix is a Lorentz transformation was observed in [68]. There the operator M = O −1 − O + was introduced; its matrix form is It is not hard to show that det Λ −1 = sdet M = sdet O + / sdet O − = 1, with the last equality following from sdet O T + = sdet O − . This guarantees that we are dealing with an SO (1,9) transformation, so the duality frame must be IIB or IIB * . Actually, it is clear that Λ a b ∈ SO + (1,9) for η sufficiently small, since it is continuously deformable to the identity; this property should hold so long as we restrict to the η locus where O ± is invertible. Then the vielbein and gravitino one-forms can be read off from (2.35) 1,9), the second gravitino is of the same chirality as the first, so we have written the above in terms of 16-component Weyl spinors. These superficially differ from the corresponding formulae in [68] in a few ways. The first is that the expressions in [68] are defined on the full group manifold rather than the physical coset. + for the latter). We similarly can read off the dilatini directly using (2.30): These agree with [68] although the intermediate expressions differ. The Ramond-Ramond bispinor can be read off from either D α or Dᾱ using and applying (B.13).
To recover the original σ-model is straightforward. It should be of Green-Schwarz form (6.11), since we have imposed the Green-Schwarz constraints. The symmetric term matches the vielbein (6.30a). The antisymmetric term is recovered by working out the B-field by comparing (6.24) with (2.38). The result is in agreement with (6.11). Note that the supergeometry does not determine the overall normalization T of the Lagrangian.

The λ-deformation
The λ-deformation [61,62] (see also [89]) was extended to AdS 5 × S 5 in [60]. Strictly speaking, this is not a deformation of the AdS 5 × S 5 superstring but rather a deformation of its non-abelian T-dual. The Lagrangian can be written 22 As with the η-deformation, the group element g lies in PSU(2, 2|4). The constant k is the level of the WZW model, and the antisymmetric operator B generates the WZW term. The Lie algebra operators O ± are given by Just as for the η deformation, the Lagrangian (6.35) can be put into GS form (6.11) with The string tension is positive for k > 0 and |λ| < 1 or k < 0 and |λ| > 1. These two parameter regions are related by taking g → g −1 .
Just as for the η-deformation, we want to recover the supergeometry of this Green-Schwarz σ-model purely from the algebra. The underlying group structure of the λ defor- In terms of these, we can build T A that satisfy the supergravity constraints, under the same simplifying assumptions as for the η-deformation: The choices for T α and Tᾱ are the most general expressions subject to the condition T α , Tβ = 0. The expressions for T a , T a , T α , and Tᾱ follow from requiring the canonical choice of the dimension zero flux tensor. The choice of T r is obvious, and T r is dictated by orthonormality. Requiring T a , T b = − T a , T b = η ab fixes the normalizations b 1 and b 2 as We find here |λ| > 1. This comes about for several related reasons -the choice of λ −1 rather than λ in (6.39), the sign choice of Killing metric for the left and right sectors, etc. The reason we keep this choice is that it better matches the explicit expressions in [68] provided we keep our coset representative (4.14) for G × G. Replacing g with g −1 (or equivalently taking m = (e, g)) and flipping λ −1 to λ would give the same expressions as [68], but now with |λ| < 1, as in the σ-model. Now we apply the generalized parallelizable space construction for G × G in section 4.2, using the coset representative (4.14). As with the η-deformation, one can introduce a dimensionful parameter v when defining the generalized supervielbein. We employ the same redefinitions (6.20) as for the η-deformation, but now subject to the normalization i . The expressions for D A are a bit more cumbersome than for the η-deformation: The construction involves the left-invariant vector fields e A t A = g −1 dg and the intrinsic As with the η deformation, we have first identified the supervielbein on the full generalized parallelizable space. Following the discussion in section 5.4, we can pass to the generalized coset by taking g = f −1 nf . However, we cannot directly apply many of the formulae from that section because of the non-trivial similarity transformation applied to the generators T A (6.39). This is in contrast to the η-deformation construction, where the triangular structure of the coset supervielbein (5.88) simplified matters. In this instance, it will be easier to proceed from scratch.
The intrinsic WZW B-field becomes, for g = f −1 nf , The WZW part lives purely on the coset, while the other term has at least one leg in the subgroup F . The upshot, while far from obvious from this perspective, is that we recover the Poláček-Siegel form with We will not show this explicitly for the other terms, although it is a worthwhile exercise. From the explicit form of the covariant derivatives, we can read off The inverses of (6.46) are The Lorentz transformation that connects E M a toĒ M a is Again, it is not hard to show det Λ −1 = sdet M = sdet O + / sdet O − = 1, which follows from sdet(Ad g) = 1. This guarantees a IIB or IIB * duality frame. The supervielbein is where we are free to use 16-component spinors because the duality frame is IIB/IIB * . Following similar steps as before, we find the dilatini and two equivalent expressions for the Ramond-Ramond bispinor Again, we can directly recover the Green-Schwarz σ-model (6.11). The vielbein E a matches the desired expression and the B-field is given by An overall factor involving the tension must be separately specified. Here it is T = |k| 4π (1 − λ −4 ) with the understanding that k should be taken to be negative and |λ| > 1.
To recover the results of [68], we should chooseb 1 = −1 andb 2 = −i. The latter choice is not technically allowed since b i should be real to ensure the Majorana condition holds. However, one can interpret this as arising from writing IIB * results in IIB conventions: this introduces factors of i for objects carryingᾱ indices (see e.g. footnote 20 of [68] or section 5 of [38]). Now the sign in (6.49) is eliminated, so that Λ a Presuming this to lie in SO + (1,9), we recover the results of [68] up to an overall Lorentz transformation. However, it is by no means obvious that this is fixed in SO + (1, 9) (or SO − (1,9)). Actually, one can show by randomly sampling elements of SU(2, 2) × SU(4) that Λ a b can lie in either connected part. Moreover, turns out to be independent of λ and determined entirely by the group element g; it in fact matches the Lorentz transformation on the coset G/F determined using Ad g as in (5.60), in remarkable contrast to the η-deformation. This surprising condition follows because the element defined in (5.60) appears always to be idempotent. 23 This seems to imply that the λ deformation is not purely fixed in either a IIB or IIB * duality frame, but that this depends on the specific group element g. This is unexpected because one might very naturally expect a IIB * duality frame since the λ-model can be understood as a deformation of the non-abelian T-dual of the AdS 5 × S 5 superstring, as argued in [68]. Certainly it is possible to find IIB backgrounds for very specific cases involving AdS n × S n factors (see e.g. [90][91][92]). It would be good to understand this point better, and whether some other factor forbids these choices of group element or invalidates the naive duality argument. 24

Analytic continuation and PL T-duality
Let us briefly comment about how the η and λ models are related [15,91,93]. As discussed in section 4.3, there exist coset representatives for G × G and G C that are straightforwardly connected by analytic continuation, and so the same holds for their generalized supervielbeins. For G C , this corresponds to a different choice of isotropic subgroup (4.56) than the one (4.48) relevant for the η deformation; in other words, the η deformation should be the Poisson-Lie dual of the analytic continuation of the λ deformation.
Of course, the generalized supervielbeins built in sections 4.2 and 4.3 carry no reference to λ or η. These parameters arose from a similarity transformation to recover the physical supervielbeins with the correct supergravity flux constraints. To understand the connection, we need only compare (6.18a) to (6.39a). Since the generators on G C map to Finally, it is worth mentioning that the η and λ σ-models (6.8) and (6.35) each involve one additional parameter corresponding with an overall normalization: these are 1/t and k π . These parameters are related to the deformation parameter of the quantum group U q (psu(2, 2|4)) governing the deformed models as q = e −κt η-deformation e iπ/k λ-deformation (6.56) for κ = 2η 1−η 2 . The analytic continuation from t to k/π can be checked at the classical level by comparing the respective Hamiltonians. For these models, we find The normalization of the Schwinger term is and captures how the parameters must change, with a factor of i coming from analytically continuing the Killing form.

Results for the dilaton
We have not yet addressed the question of whether these supergravity backgrounds admit a dilaton. It was shown in [68] that the λ-deformation always admits a dilaton while the η-deformation admits a dilaton only when a certain unimodularity condition on the Rmatrix is satisfied. We can now see how these conditions arise naturally within double field theory. As discussed in section 3.2, one can replace ∂ M log Φ in the dilatonic flux tensor by a vector X M (3.33) and impose the same constraints on this flux as in super DFT [38]. This implies no additional constraints on the supergeometry: the vector X M is the DFT analogue of X M and K M in generalized supergravity. The constraints in question amount to fixing From these expressions, one can compute X α . The question is whether that can be written as D α of some superfield. Rather than compute this directly for the models in question, we will follow a less direct but more rewarding route, and address the full set of dilatonic fluxes in one fell swoop. The crucial point is that the covariant dilatonic torsions all vanish when the constraint (6.59) and the Bianchi identities are imposed [38]. These differ from the fluxes F A by the Ω connection of type II DFT, which is composed of not only the double Lorentz connection but also connections associated with the additional parameters given in Table 1.
What exactly are these Ω? Recall that the Poláček-Siegel framework furnished us a Lorentz spin connection where Ω M r was a piece of the megavielbein. Is this the right one? That question is easy enough to answer. At dimension 1/2, choosing the DFT torsion tensors T αbc and Tᾱ bc to vanish fixed the α component of Ω. Indeed, we can check that (similarly for the barred versions) where F αbc is the flux for the megavielbein (which vanishes for both cases of interest). The other dimension 1/2 torsion tensors T αβ γ , T αβ γ , T αβ γ , and their barred versions similarly match the corresponding generalized flux tensors (all also vanishing). At dimension 1, we find implying that Ω [abc] and Ω abc and their barred versions are chosen properly. At dimension 1 we also have Both of these should vanish. Since Fᾱ b γ ∝ κ γγ (γ b )γᾱ is γ-traceless, using the properties of κ αᾱ , there is no obstruction to choosing Ωᾱ b γ = − Fᾱ b γ so that first torsion vanishes.
The second vanishes since F αb γ = 0 and so we can choose Ω αb γ = 0. At dimension 3/2, we have All the generalized flux tensors vanish on the right, and so we are free to choose all the corresponding Ω's to vanish. 25 What does this mean for T A ? From the conditions derived on the non-Lorentz Ω, we find (6.66) 25 Strictly speaking, we can only fix Ω up to the residual shift symmetries discussed in [38].
Each of these can be interpreted as pieces of the dilaton flux tensor on the Poláček-Siegel megaspace (5.38). We know for a supergravity solution, all of these must vanish. Moreover, from the dilatonic Bianchi identity, we also know that the dilatonic SO(4, 1) × SO(5) curvature R ab = −R r f rab vanishes. The upshot is from (5.33) we can impose the strictest possible condition on the Poláček-Siegel dilatonic flux, with the vanishing of the second term following from the properties of PSU(2, 2|4). This means that for both the η and λ deformations, the generalized dilatonic torsion in the Poláček-Siegel framework must be taken to vanish, F A = 0. The results in section 4.4 apply for F A = F A = 0. For G × G, we have from (4.62) To see this, one first exploits where the • denotes an irrelevant quantity. From this, we can immediately see respectively, and the superdeterminants of these are just ±1. A similar line of argument establishes that sdet O − is proportional to sdet O + , and these are also proportional to the full operators O ± . This recovers the result of [68].
For G C , we first observe from (4.64) that Therefore, the existence of a dilaton solution requires the unimodularity condition for the R-matrix, R B C f C B A = 0. Provided this holds, we recover the same conditions, and an identical line of reasoning leads to (6.71) for the corresponding operators O ± . This again is in full agreement with [68].

Discussion
In this paper we have discussed how to employ superspace double field theory, involving a generalized supervielbein, an element of OSp(D, D|2s), to describe generalized dualities. We confirmed our initial expectation that all algebraic structures relevant for dualities of the bosonic string carry over to generalized supergeometry naturally. When the generalized flux tensor is constant, the space is generalized parallelizable (or a generalized coset thereof), and one can construct the generalized supervielbein explicitly in terms of the group theoretic data. A considerable advantage is that the generalized supervielbein unifies all fields of type II supergravity, except for the dilaton, in one object. To appreciate this fact, recall the salient features of established generalized geometries for type II strings: • In O(D, D) generalized geometry, the metric and B-field are unified by the generalized frame, while the Ramond-Ramond sector can be captured either with an O(D, D) Majorana-Weyl spinor [34,35] or an O(D − 1, 1) × O(1, D − 1) bispinor [36,37] (see [73] for the relation between them). The Ramond-Ramond sector and the generalized frame are a priori independent objects, related only by the field equations.
• Exceptional generalized geometry improves the situation by incorporating the Ramond-Ramond sector into the generalized frame. However, this requires the transition from a T-duality covariant description to a U-duality covariant one. Consequentially, strings are no longer the fundamental objects. They are replaced by membranes, which come with their own challenges. When the full ten-dimensional spacetime needs a unified treatment, like for the η and λ-deformations of the AdS 5 × S 5 superstring, one has to deal with the infinite dimensional duality group E 11 (11) [94][95][96] which is not completely understood yet (see [97][98][99] for recent progress).
Additionally, neither approach directly incorporates fermionic dualities. All these problems are resolved by generalized supergeometry making it the ideal framework to analyze integrable deformations of superstrings. Therefore, one main focus of our efforts was to explain the η and λ deformations within superspace double field theory. While their σ-model actions are fairly complicated, their explanation within super-DFT is rather straightforward, in terms of the double Lie groups G × G and G C , with a single parameter (η and λ, respectively) describing how the supergravity frame is embedded in the doubled space.
dim. constraint A major novelty compared to the purely bosonic approach is the necessity of additional torsion constraints, which restrict the generalized fluxes beyond their Bianchi identities. They fix the form of their dimension − 1 2 and dimension 0 components as in Table 2: these imply similar constraints in generalized type II supergravity [77]. From the worldsheet perspective, these are required for the underlying Green-Schwarz superstring to possess κ-symmetry. Consequentially, the target space supergeometry satisfies the field equations of generalized supergravity [77,78]. Moreover, they put the theory on-shell; otherwise, supersymmetry transformations would not close into an algebra.
As one can see from Table 2, these flux constraints are not covariant under OSp(D, D|2s) transformations. Rather, they break the duality group to the local symmetry group H L × H R , which plays the same role as the double Lorentz group in bosonic DFT. In the latter, the generalized metric is responsible for the breaking. Due to the absence of a generalized supermetric in the supersymmetric extension, the flux constraints take over this function, too. This is analogous to the situation in conventional supergravity: there the torsion constraints are essential and there is no Riemannian supermetric.
There are several additional avenues one could explore at this point. One issue we avoided discussing was the σ-model interpretation of generalized dualities. These are described in terms of the E-model [13][14][15] and its dressing coset extension [19,100]. These models can be straightforwardly built for supergroups, but a subtlety involves finding the right constraints to ensure that the σ-model is of Green-Schwarz form. This would undoubtedly be related to a duality-symmetric formulation of the GS superstring using the language of super-DFT [38].
Another avenue to explore is the potential connection with integrability. The η and λ deformations were initially constructed as integrable deformations of the AdS 5 × S 5 superstring, and a key role is played by the Z 4 grading in the supergroup. It is already known that there are connections between the structure of E-models and integrability [70,71]. It would be interesting to explore the connection for the case of super E-models.
Generalized dualities have proven to be useful solution generating techniques. Examples include non-abelian T-duals of backgrounds like AdS 5 × S 5 and AdS 3 × S 3 × T 4 [46] which are relevant for the AdS/CFT correspondence. In this context, an important question is how much supersymmetry of the original background is preserved by T-duality. In our framework the amount of supersymmetry is fixed by the number of fermionic generalized Killing vectors. Therefore, one should study how they transform under duality transformations. Perhaps one could construct a systematic treatment within super-DFT. One could then revisit known examples and try to exhaust all possible dualities to find new solutions.
Finally, we should add that very significant work on U-duality extensions of Poisson-Lie T-duality and its generalizations has appeared recently [101][102][103][104][105][106]. These would undoubtedly have natural descriptions in supersymmetric extensions of U-dual formulations, of the type explored e.g. in [98,107,108].
Both the algebra and the pairing can expressed purely in terms of the generators t A , but it depends on whether the generators t A are treated as commuting quantities, ξ α t α = t α ξ α or as formal graded objects themselves, ξ α t α = −t α ξ α . The first situation applies when the superalgebra g is embedded in a supermatrix algebra gl(m|n); in this case, the generators themselves are matrices of (commuting) complex numbers, and (A.1) and (A.3) imply The second situation, where the t A are themselves graded, leads to the more conventional expressions where gradings arise primarily because of index ordering and the direction of contraction. We will employ the latter conventions when explicit indices are exhibited. The sign convention for f AB C is a bit unconventional; this is to ensure the torsion tensors for supergroup manifolds have a plus sign, i.e. T AB C = +f AB C .

A.2 The orthosymplectic group OSp(D, D|2s)
An element of OSp(D, D|2s) is described by a graded supermatrix U M N ∈ GL(2D|2s) satisfying the condition for a graded symmetric matrix η MN with graded inverse η MN , It can be naturally described in terms of its GL(D|s) subgroup where a generalized vector V M decomposes as a one-form and vector V M = (V M , V M ). In this basis, η is given by Because of the grading present in η, it matters whether an index is raised or lowered. We conventionally identify elements of a matrix U M N as if they were elements of U MN , i.e.  Table 3. Classification of type II duality frame.

B Democratic Type II supergravity conventions
We summarize here our conventions for democratic type II supergravity and how they arise from DFT. Conventions for 10D gamma matrices and spinors can be found in [38]. The inspiration for such a "democratic" approach to type II was inspired by Wulff, see the appendices of [109]. The supervielbein emerging from DFT consists of two copies of the vielbein super one-form E M a and E M a , as well as two gravitino super one-forms, E M α and E Mᾱ . The two vielbeins are related by a Lorentz transformation that determines the duality frame relative to IIB. That is, Λ a b is an element of O (α,β) (1,9), where α Λ = −1 or β Λ = −1 if Λ involves a temporal or spatial orientation reversal, and +1 otherwise, see Table 3. We may think of Λ a b as a similarity transformation to convert barred vector indices to unbarred ones. In order to convert barred spinors to unbarred ones, we introduce the spinorial matrix / Λ = (Λαβ), which obeys The last condition implies that / The supercharges Q iα obey analogous formulae as the dilatini and satisfy the SUSY algebra where we use the chiral projector P L = 1 2 (1 + γ * ). The second SUSY involves a projector P L = 1 2 (1 + α Λ β Λ γ * ), which is P L for IIB/IIB * and P R for IIA/IIA * .

(B.13)
This S differs from [77,109] by a factor of −16i. An extra factor of two comes from employing the democratic formulation with both field strengths and their duals. Employing 32-component Majorana spinors can be inconvenient when exhibiting the various torsion tensors. This was addressed in [38] by introducing tilde spinors for the second copy of the gravitini and dilatini so that 16-component Majorana-Weyl notation can be used throughout. Effectively, tilde spinors are just barred spinors of DFT, reinterpreted as either same chirality or opposite chirality as unbarred spinors, depending on the duality frame, i.e. E Mα is E M 2α δ αα or E M 2 α δ αα . We do not employ tilde spinors in the main body of this paper, but they are convenient for describing the superspace curvatures without sprinkling chiral projectors everywhere. First, we introduce tilde γ matrices as In terms of these, the non-vanishing torsion tensors are given through dimension 1 by T γβ α = 2 χ (γ δ β) α − (γ a ) γβ (γ a χ) α , Tγβα = 2 χ (γ δβ )α − (γ a )γβ(γ a χ)α , (B.16b) The dilatini χ α and χα are given by the spinor derivatives of the dilaton C Gauged superspace σ-models In this appendix, we provide a concise extension of the work of Hull and Spence [110,111] to superspace (see also [112] and [113]). In large part, this is merely a relabeling of indices and the addition of a grading, but we include it here for the reader's convenience.

C.1 Target space supergeometry
A superspace σ-model comes equipped with a graded symmetric rank-two tensor G M N and a super two-form B M N . We presume there exist certain superisometries k R = k R M ∂ M , which leave G M N and H = dB invariant. The latter condition means that for some one-form v R = dZ M v M R . We use the convenient shorthand L R ≡ L k R . The Killing supervectors k R obey the algebra [k R , k S ] = f RS T k T . The following conditions hold: where we introduce a locally defined, (graded) antisymmetric scalar function Λ RS and the (graded) antisymmetric constant c RST . As a consequence of the above equations, one can show 3) The closed one-form w introduced in [110] corresponds to d(k R v S − Λ RS ) here.
There is some gauge redundancy in these quantities: where c RS is an antisymmetric constant and ρ R (Z) is a scalar function of the target space coordinates, with a residual "gauge-for-gauge symmetry" of δρ R = c R and δc RS = −f RS T c T . One can also define the isometry on the B field directly, L R B = d v R + k R B = −dω R . Then in the context of generalized geometry, one can speak of the generalized vector ξ R = k R + ω R with Dorfman bracket [ξ R , ξ S ] D = f RS T ξ T + d(Λ RS − k R v S ) obeying a generalization of the non-abelian algebra of the k R . The additional one-form term above is a trivial transformation from the perspective of double field theory.

C.2 Gauged σ-model
The ungauged σ-model is given as the sum of a kinetic and Wess-Zumino term, It possesses a global symmetry δZ M = λ R k R M . To gauge it, we introduce a worldsheet one-form A R that transforms as so that DZ M := dZ M + A R k R M transforms as δDZ M = λ R DZ N ∂ N k R M . The kinetic term is then invariant by simply replacing dZ M → DZ M . The Wess-Zumino term is more involved. Let us simply give the answer: Pullbacks to the worldsheet are implicitly assumed in the above equations. In the final term, we have used the field strength and included a Lagrange multiplier χ R whose equation of motion enforces that A R is pure gauge. Strictly speaking the gauged σ-model lacks the Lagrange multiplier term, but we will include it since we are interested in performing a duality transformation.
In order for the Wess-Zumino term to be invariant, we must impose two conditions. First, the Lagrange multiplier field χ R must transform as With this condition, the Wess-Zumino term varies (up to a total derivative) into and so invariance actually requires this constant to vanish, c TSR = 0 . This is a crucial consistency condition for the ability to gauge the action. The Lagrange multiplier field must transform under the residual symmetry (C.4) as δχ R = −ρ R (Z). The residual constant c RS shift is no longer a symmetry of the action: it instead leads to different gauged actions whose Λ RS factors differ by such a constant. Because c RST vanishes, such shifts must obey the cocycle condition f [RS| U c U|T] = 0. In a standard gauging, the Lagrange multiplier is absent and so one must be able to consistently fix χ R = 0, leading to This is a key condition discussed in [110]. It turns out that a consequence of (C.11) is that c RST vanishes, so we can consider the former condition as fundamental. Once the condition (C.11) is imposed, the residual symmetry parameter ρ R in (C.4) is restricted to obey L R ρ S = f RS T ρ T . In principle, the duality can proceed directly by integrating out the gauge fields A R . The resulting action admits the local λ R gauge symmetry, implying that dim G coordinates are unphysical and can be eliminated by a gauge-fixing. A simpler procedure is to go to adapted coordinates.

C.3 Adapted coordinates
If the isometries act freely, one can select out dim G coordinates so that k R = k RṀ ∂Ṁ . In these adapted coordinates Z M = (Z M , YṀ ) where Z M are spectator coordinates. We do not address the non-free case, but one can follow a very similar line of reasoning.
Let g(Y ) be a group element for the group G we are gauging. The left and rightinvariant one-forms are e R t R = g −1 dg and k R t R = dgg −1 with the generators obeying (3.3). The Killing vectors k R obey k R k S = δ R S and k R e S = (Ad g −1 ) R S . We definẽ A R := DZṀ eṀ R = dZṀ eṀ R + A S k SṀ eṀ R = (k S + A S )(Ad g −1 ) S R .

(C.12)
This is a gauge-invariant one-form, δ λÃ R = 0. The kinetic term can be written where we have flattened theṀ indices on the metric with e RṀ . Every piece above is separately gauge invariant. For the metric, the invariance condition reduces to independence of YṀ . The Wess-Zumino term is more involved. First, trade A R forÃ R . The result is structurally identical and reads where the tilded quantities are defined as The tilded quantitiesṽ R andΛ SR obey the useful relations: The right-hand sides of both these expressions are annihilated by e R , so they are independent of dYṀ . The field strengthF R can be expanded out to rewrite the Wess-Zumino term as In this form, it's very easy to show gauge invariance of the second and third terms using To understand the meaning ofB, it helps to rewrite it as If we want the λ gauge symmetry to be completely eliminated at this stage, so that e.g.χ R is invariant, we should chooseṽ SR = 0, which is the consistency condition (C.11) discussed earlier. A consequence of this condition is that 25) This means that the Wess-Zumino term can finally be written as 26) in terms of the original B-field. The components of B in the second line above, are each independent of the coordinate YṀ . Relabelingχ R as ν R , we recover the recipe for gauging reviewed in section (3.1).

D Flux tensors for η and λ deformations
We summarize the structure constants F A B C relevant for the η and λ deformations below by their dimension. Both cases can be given in terms of coefficients c 1 and c 2 (which are proportional to a i or b i ) as well as a function Γ.
Dimension 0 The coefficients c i appear in the fluxes only quadratically. In terms of the generators T A given in sections 6.2 and 6.3, the coefficients become c i c j = a i a j (1 + η 2 ) = b i b j λ −1 (D.1) Specifying the coefficients quadratically circumvents introducing a square root. One can check that the two expressions for c i c j go into each other under the analytic continuation (6.55). The function Γ is given in the two cases by For the η deformation, |Γ| ≤ 1 for all values of η and vanishes at |η| = √ 2 ± 1. For the λ deformation, Γ ≥ 1 and saturates the lower bound at λ = 1. The highest dimension structure constant F rst involves For reference, we also give some of the relations above in terms of κ = 2η 1−η 2 : (D.4) Upon truncation to the bosonic sector, it is κ and λ 2 that play the role of the parameters for the conventional η and λ deformations for a group G.
After the redefinitions to go the supergravity frame, the derivatives D A in (6.24) and (6.42) have flux tensors with F A B C formally given by the F A B C above, but with the replacements whereâ i andb i denote the phases of those quantities. In section 6.2, we choseâ 1 =â 2 = 1.