Generalized Dualities and Higher Derivatives

Generalized dualities had an intriguing incursion into Double Field Theory (DFT) in terms of local $O(d,d)$ transformations. We review this idea and use the higher derivative formulation of DFT to compute the first order corrections to generalized dualities. Our main result is a unified expression that can be easily specified to any generalized T-duality (Abelian, non-Abelian, Poisson-Lie, etc.) or deformations such as Yang-Baxter, in any of the theories captured by the bi-parametric deformation (bosonic, heterotic strings and HSZ theory), in any supergravity scheme related by field redefinitions. The prescription allows further extensions to higher orders. As a check we recover some previously known particular examples.


Introduction
Probing space-time with strings challenges the way we describe geometry. When the target space possess commuting isometries string theory is invariant under Abelian T-duality, the statement that different backgrounds lead to the same underlying physics. Double Field Theory (DFT) [1,2] is a framework that accounts for such a generalized geometry by making Abelian T-duality a manifest symmetry, for reviews see [3]. physics 2 . This observation was originally done in [13], and lies at the core of many interesting discussions on how DFT connects with generalized dualities [15]- [20]. We can resume it as follows: Generalized dualities are represented through certain local O(d, d) transformations and shifts of the generalized dilaton that relate different backgrounds (duality twists in Gauged DFT) whose gaugings fall into the same duality orbit.
In this paper we exploit the technology of DFT to compute the first order higher derivative corrections to generalized dualities. Higher derivatives are incorporated into DFT through deformations of the double local Lorentz transformations [21]- [22] 3 . Connecting the duality covariant DFT fields with those of supergravity requires both a Lorentz gauge fixing and certain higher order field redefinitions. While the original DFT fields transform covariantly under O(d, d), the T-duality transformation of the supergravity fields gets deformed after the gauge fixing due to the compensating Lorentz transformations and the redefinitions. Interestingly, throughout this procedure the O(d, d) transformations need not be rigid, and so it can be applied to generalized dualities in light of the observation made above.
In this paper we find a unified expression for first order corrections to generalized dualities. It can be easily specified to any generalized T-duality (Abelian, non-Abelian, Poisson-Lie, etc.) and deformations such as Yang-Baxter, in any of the theories captured by the higher derivative corrections to DFT (bosonic or heterotic strings and HSZ theory), in any supergravity frame related by field redefinitions.
Before introducing the original results, we intend to provide a pedagogical introduction to DFT for readers of the generalized duality community, and the other way around. Section 2 is devoted to review some relevant aspects of DFT, its flux formulation, its gauged version and the way to encode higher derivatives. We discuss there how generalized dualities fit into Gauged DFT to leading order in α ′ . Section 3 discusses how generalized dualities are captured by local O(d, d) × R + transformations, and present the explicit form of the elements of this group in the case of Abelian, non-Abelian and Poisson-Lie T-duality. Section 4 contains most of the original results of this paper, combining the ideas in section 2 and 3 to generate a general formula for higher derivative corrections to generalized dualities.
Along the paper the reader will find the following results: • Although local O(d, d) transformations and R + shifts of the generalized dilaton are not symmetries of DFT, some specific elements of this group transform Gauged DFT into another Gauged DFT in the same duality orbit. In certain cases when the gaugings fall into distinct duality orbits, one can still make sense of the transformation as connecting background solutions to deformed DFTs.
• The local O(d, d) × R + transformations that relate dual backgrounds remain uncorrected with respect to higher derivatives. The elements that generate the generalized dualities can then be read from the backgrounds to lowest order, and applied to higher-order corrected backgrounds so as to obtain the corrections to the dual background. This result is extremely powerful, as it allows to perform duality transformations to backgrounds with higher derivatives, by knowing only the transformation to lowest order. We give the explicit form of these transformations for different generalized dualities: Abelian (3.49), non-Abelian (3.59) and PL T-duality (3.96), and discuss the relation between pluralities and the notion of orbits in Gauged DFT.
• In the context of Gauged DFT, the duality covariant generalized fields are linearly acted on by the local O(d, d) × R + transformation that defines the generalized duality. However, when it comes to translating this into the language of supergravity, the gauge fixing and field redefinitions spoil the order and simplicity of generalized dualities in Gauged DFT, inducing higher-order corrections to the transformations of the supergravity fields. In this paper we compute these corrections in full generality in (4.29). The result is remarkably simple, and still general enough to account for any of the two parameters a and b that control the higherorder deformations (a = 0 or b = 0 is the heterotic string, a = b is bosonic, a = −b is HSZ), for any generalized duality (defined as connecting background solutions through local O(d, d) transformations and generalized dilaton shifts), for any supergravity scheme defined by its relation to the DFT scheme.

A review of Double Field Theory
In this section we set the conventions to be used throughout the paper, and briefly review the frame [1], [25] or flux [26] formulation of DFT, it's gauged version [12] through gSS reductions and it's first order higher-derivative extension [21].
We begin with some conventions. D is the dimension of the full space-time, d is the dimension of the internal compact space, and n = D − d is the dimension of the external space. Apart from the usual curved and flat type of indices, flux compactifications involve an extra type of internal indices that we call "algebraic" for reason that will become clear later. Table 1

Flux formulation of DFT
Double Field Theory (DFT) incorporates T-duality as a manifest symmetry, given by the continuous global O(D, D) group that preserves the symmetric metric η MN . This metric and its inverse are used to raise a lower the 2D curved indices M, N on which O(D, D) acts. Duality requires that in addition to the standard space-time coordinates X µ , the theory includes dual coordinates X µ , associated with the winding excitations of closed string theory on backgrounds with non-trivial cycles. It is then defined over a doubled space with coordinates X M = (X µ , X µ ). The double space is however constrained. One option is to impose the strong constraint which states that all fields and their products must be annihilated by the double Laplacian This implies that locally there is always an O(D, D) transformation that rotates into a frame in which the fields depend only on half of the coordinates. A particular solution is given by demanding that nothing depends on the dual coordinates ∂ µ = 0 in which case the section coincides with the standard D-dimensional space-time on which supergravity is defined. Flux compactifications of DFT permit a relaxation of this strong constraint [12], as we will discuss later.
There is also a local O(1, D − 1) × O(1, D − 1) symmetry usually referred to as the double Lorentz symmetry. It preserves two symmetric matrices η AB and H AB and acts on flat 2D indices A, B which are raised and lowered by η AB .
The field content of the theory simply consists of a generalized frame E M A and a generalized dilaton d, that depend on the double coordinates. The generalized frame is constrained to satisfy and permits to define the famous generalized metric as follows 3) The O(D, D) transformations act linearly on the space and fields through matrix multiplication The double Lorentz transformations act on the fields as follows It is convenient to define a different set of double Lorentz invariants which are projectors P (±)2 = P (±) and P (±) P (∓) = 0 acting on the different factors of the double Lorentz product. We define the following index notation for future reference On top of these symmetries, DFT is invariant under generalized diffeomorphisms, which will play a minor role in this work. Finally, there is a crucial transformation consisting in a constant generalized dilaton shift, that we will call R + This is not a strict symmetry of the action, but a rescalling, and then the equations of motion turn out to be invariant under this symmetry. This will be crucial when it comes to gauging the theory.
DFT is defined by an action that is fixed by invariance under the symmetries discussed so far. In the frame formulation, it can be written compactly in terms of the so called generalized fluxes (2.12) The specific form of the action and the corresponding equations of motion are irrelevant in this paper, the only important thing we need to keep in mind is that they can be written in terms of the generalized fluxes and their flat derivatives (see [26] for the two-derivative action, and [22] for the first order corrections in terms of fluxes). Of special importance are certain projections of the generalized fluxes that happen to appear in higher derivative Lorentz transformations, and so we define them here for future reference Connecting with supergravity requires a GL(n) × O(d, d) decomposition of O(D, D). Let us show how this works in the fully uncompactified scenario n = D. We first impose the strong constraint and pick the solution ∂ µ = 0, so nothing depends on the dual coordinates. Next, we propose a parameterization of the generalized frame and dilaton and also the invariant matrices Here Q µν ≡ G µν + B µν and g (±) αβ = diag{−1, 1, . . . , 1} are D-dimensional Minkowski matrices that raise and lower flat D-dimensional indices. There are two vielbeins e (±) µ α each transforming under different factors of the Lorentz group. They differ by a Lorentz transformation, and so they generate the same metric If desired, the generalized metric can then be computed from these definitions (2.3) Using the parameterization of the generalized fields we can compute the components of the generalized fluxes. In particular we show here the non-vanishing components of F (±) in (2.13) where ω(e (±) ) and H(e (±) ) are the Levi-Civita spin connection and curvature for the two-form respectively but evaluated in e (±) instead.
Making contact with supergravity requires a gauge fixing. This can be achieved by taking (2.20) On the one hand this gauge fixing breaks the double Lorenz group down to its diagonal subgroup, and on the other it breaks the O(D, D) covariance of the generalized frame, so the failure of O(D, D) to preserve the form of the generalized frame after the gauge fixing will have to be compensated by a restoring double Lorentz transformation. We will discuss this extensively later.

Gauged DFT
We now briefly review Gauged DFT [11]- [12], which is obtained after performing a generalized Scherk-Schwarz (gSS) reduction [10] of DFT. The idea is to keep the O(D, D) structure of the theory, assuming an underlying GL(n) × O(d, d) decomposition, under which the coordinates split as X M = (X m , X m , Y M ) and the strong constraint is imposed in the external space such that ∂ m = 0. The gSS ansatz for the fields is read from the rigid O(D, D) × R + symmetries of the equations of motion, and separating the dependence on external X and internal Y coordinates where the fields with a hat only depend on the external coordinates and correspond to the dynamical objects in Gauged DFT. The matrix U (Y ) is usually called twist matrix or duality twist, as it must be O(D, D) valued. It maps indices of the parent DFT M, N to indices of the effective Gauged DFT I, J , and must be trivial in the external directions so it is in fact an element of O(d, d). Together with λ(Y ), they encode all the dependence on the double internal coordinates, and contain the information of the compactification background.
To understand the physics behind the ansatz, it is instructive to see how it affects the generalized metric The full background H(X, Y ) MN is written as perturbations around the compactification back- where the fluctuations are governed by H(X) IJ around δ IJ , which contains the fields in the effective action of Gauged DFT, and is fixed by it's equations of motion.
Under the gSS ansatz the generalized fluxes (2.12) split as a sum of external and internal parts where all the dependence on the twists ends on the gaugings, defined by (2.25) Invariance of the action, covariance of the equations of motion and closure of the gauge algebra leads to a set of consistency constraints where we are defining ∂ I = U M I ∂ M . Interestingly, the strong constraint implies these equations, but the reserse it not true and so this is a relaxed version of the strong constraint in the internal space, which can be truly double as long as these quadratic constraints are satisfied [12]. Normally, the gaugings F I receive extra contributions through the gauging of a warp factor re-scaling of the Kaluza-Klein fields that arise under a GL(n) × O(d, d) decomposition. We are not assuming such a decomposition and so we will ignore this here, for a general discussion on this point we refer to [10] and [27]. We finally point out that normally the fluxes are taken to be constant, in which case the action reduces to a lower dimensional gauged supergravity. Here we will not always assume this, as non-constant deformations are relevant when it comes to discuss certain backgrounds that arise in the context of generalized dualities.
Since the twist matrix has to be trivial in the external sector (2.22) it can be parameterized as that satisfy their own Jacobi identities In the effective action, all the information of the background is encoded exclusively in the gaugings F IJK and F I . Their explicit form will depend on the twist matrix U , which in full generality is given by [26] The so called geometric and non-geometric fluxes [28] in this context are simply particular components of the gaugings, and can be expressed in terms of these background fields [10] A priori there is no obstruction in the formalism to reach all possible orbits of gaugings (this was proved for O (3,3) in [13]) if the strong constraint is relaxed as in [12], although a proof is still missing in general. It was shown in [13] that when the twists are strong constrained, they additionally satisfy which is the condition that the gaugings admit an embedding into maximal supergravity [29]. This is not a constraint of Gauged DFT. Only a subset of the allowed gaugings satisfy this condition, and so a relaxation of the strong constraint is mandatory in order to reach all duality orbits. We refer to [13] for discussions on this point.
Let us discuss the idea of how generalized dualities are treated in the context of Gauged DFT. Consider a background coordinatized by Y and characterized by U (Y ) and λ(Y ) with gaugings Next consider a different (dual) background coordinatized by Y ′ and characterized by U ′ (Y ′ ) and When the gaugings fall into the same duality orbit, namely when there exists a constant element The combined action of (2.35) and (2.36) leave the full generalized fluxes (2.24) invariant Moreover, since these fluxes only depend on the external coordinates X, flat derivatives acting on them are also invariant under this transformation (D A F ) ′ = D A F . As a result, the full Gauged DFT action and it's equations of motion remain invariant. The resulting effective theory for both dual backgrounds is the same, and in this sense they are dual to each other. Moreover, if the external factors of the gSS ansatz E(X) and d(X) satisfy the equations of motion of Gauged DFT, then the generalized duality maps a solution to a solution. It is then trivial from the point of view of Gauged DFT that generalized dualities act as a solution generating technique at the classical level. This is nicely discussed in [18].
The twists and their duals belong to different spaces with different set of coordinates, Y for the original and Y ′ for the dual. We can think of going from one background to the other through a transformation 4 consisting in specific local O(d, d) rotations by the elements ψ(Y, Y ′ ) and local generalized dilaton that connect backgrounds whose gaugings fall into the same duality orbit. It is in this sense that generalized dualities can be defined by promoting the global symmetries of DFT into local symmetries of Gauged DFT.
We can summarize how generalized dualities are captured by Gauged DFT as follows: Although local O(d, d) transformations and R + shifts of the generalized dilaton are not symmetries of DFT, some specific elements of this group transform Gauged DFT into another Gauged DFT in the same duality orbit. Now suppose the following scenario. We have a local O(d, d) × R + transformation connecting two backgrounds (U , λ) and (U ′ , λ ′ ) that generate gaugings that fall into distinct duality orbits. In this case, it might be possible to deform them (by modifying the twists) and force them to coincide. If the deformation on its own generates a consistent gauging, then the backgrounds can be interpreted as solutions to different Gauged DFTs gauged by the deformations. We will see this effect explicitly when discussing particular examples of generalized dualities.
The local O(d, d) × R + transformations that connect twists (U , λ) and (U ′ , λ ′ ) that fall into distinct duality orbits, can sometimes be interpreted as a mapping between solutions of deformed theories.

Higher derivatives in DFT
In this section we review how to incorporate higher-derivatives in DFT through corrections to the double Lorentz transformations [21]. The infinitesimal first-order in α ′ deformation is given by the generalized Green-Schwarz transformation (antisymmetrization of projected indices exchanges the index but not the projection where the overline indicates that the components are duality covariant but not Lorentz covariant.
In other words, the duality covariant fieldsΨ are related to the Lorentz covariant ones Ψ though first order redefinitions ∆Ψ, namelyΨ = Ψ + ∆Ψ. Note thatΨ is duality covariant but Lorentz non-covariant, and Ψ is the opposite. The parameterization of the first-order deformation is (2.42) Note that because this deformation is already first-order, it is the same to put bars or not as the difference is of higher order. The corrected transformations of the D-dimensional fields are given by where we have written everything in matrix notation.
As we briefly mentioned before, and will extensively explain later, when it comes to reduce this setup to supergravity a gauge fixing is required. Interestingly, this gauge fixing breaks the O(D, D) covariance of the theory, and then compensating double Lorentz transformations are required to restore the gauge. Since the elements of O(D, D) are finite, so must the double Lorentz transformations. We have discussed only infinitesimal corrections to double Lorentz transformations through the generalized Green-Schwarz deformations (2.40), but when it comes to the gauge fixing this is not enough: finite compensating double Lorentz transformations are required to restore the gauge broken by finite O(D, D) transformations. So, let us then discuss how to extract the finite Lorentz deformations from the infinitesimal ones considered above, following the strategy in [30] closely. We aim at re-writing the transformations in terms ofŌ (±) = 1 +Λ (±) + . . . where the dots represent higher orders inΛ (±) , such thatŌ (±) g (±)Ō(±)t = g (±) . Since the lowest order is trivial, let us focus on the generalized Green-Schwarz transformation. To this end, consider the finite and infinitesimal transformation of the spin connections (which follows from L(ē (±) ) =ē (±) O (±) ) Using the above we take the following tour for the symmetric part of Σ (±) in (2.42) What we did above is the following. We identified Σ (±) (µν) with the infinitesimal failure of 1 2 tr(ω (±) ω (±) ) to remain invariant, and the arrow indicates that we now replace Σ (±) (µν) by the failure of 1 2 tr(ω (±) ω (±) ) to be invariant under finite Lorentz transformations.
For the antisymmetric part of Σ (±) we proceed similarly. First we note thatB µν recieves a first order Lorentz transformation from the generalized Green-Schwarz term, given by δ ΛBµν = −2Σ [µν] , which implies that H µνρ = 3∂ [µBνρ] cannot be the three-form field strength as it is not Lorentz invariant The failure coincides with the infinitesimal Lorentz transformation of two copies of Chern-Simons three forms As before, we now consider the finite Lorentz transformation of the Chern-Simons three-forms and considering that the last term is closed and then locally exact, we readily arrive at In conclusion, the finite version of the generalized Green-Schwarz transformation on D-dimensional fields is as follows We have also included the Lorentz transformation for the dilaton field which is obtained from L(d) = d and its parameterization (2.41). This result is completely general, it holds for any choice of the parameters a and b and no gauge fixing was assumed.
On a different page, let us comment here how this setup can be used to compute higher derivative corrections to generalized dualities. To address this question we must follow the approach in [22], which is simply the gauged version of the α ′ deformed DFT [21]. The idea is to perform a gSS reduction of DFT to first order in α ′ , which interestingly proceeds in exactly the same way as in the two-derivative case. When the gSS ansatz (2.21) is adopted, the twists U (Y ) and λ(Y ) end up forming the exact same fluxes that gauge the action, equations of motion and gauge transformations in the two derivative action. This is, nor the twists nor the gaugings receive higher-derivative corrections. However Lorentz transformations. In the notation adopted so far, these components carry a bar to indicate that they are not supergravity fields. As before, a gauge fixing plus Lorentz non-covariant field redefintions are required to relate them to the standard fields in supergravity. These redefinitions are first-order in α ′ , and so they induce higher derivative corrections in E(X) and d(X) (2.55) Following the logic of how generalized dualities are captured by Gauged DFT, we can now perform the local O(d, d) transformations and shifts of the generalized dilaton (2.38) and (2.39) to transform the background into its dual (2.56) As before, when the dual background U ′ (Y ′ ) and λ ′ (Y ′ ) generates gaugings that fall into the same duality orbit than those of the original background, then it is guaranteed to be a solution of the α ′ corrected DFT. If instead the orbits are different, the dual background could be a solution of a deformed α ′ corrected DFT.
There is a remarkable consequence of the fact that the twists receive no corrections and that all the corrections are captured by the external part of the gSS ansatz: The local O(d, d) × R + transformations that relate dual backgrounds remain uncorrected with respect to higher derivatives. Then, we can read the elements ψ(Y, from the backgrounds to lowest order, and apply the transformation to higher-order corrected backgrounds so as to obtain the corrections of the dual background. In the context of Gauged DFT, the duality covariant generalized fields are simply acted on linearly by the local O(d, d) × R + transformation that defines the generalized duality. When it comes to make contact with supergravity, the gauge fixing and required field redefinitions will induce higher-order corrections to the generalized dualities. In this paper we will compute the first-order corrections in full generality.

O(D, D) structure of generalized dualities
We have defined generalized dualities as the combined action of specific local O(d, d) transformations and generalized dilaton shifts R + that map solutions into solutions of Gauged DFTs. In this section we present the explicit form of these elements for the cases of Abelian, non-Abelian and PL T-dualities, and also Yang-Baxter deformations. In addition we discuss the embedding of these dualities into the full O(D, D) × R + .

Decompositions of O(D, D)
We introduce here how to decompose the group O(D, D) into its subgroups GL(D) (useful to deal with full D-dimensional solutions) and GL(n) × O(d, d) (more relevant in compactification scenarios).

GL(D) decomposition
We now review the aspects of O(D, D) that will be relevant to us, for more details see [31]. The O(D, D) group can be spanned by the matrices where the bullets represent the index structure and the D × D matrices have to satisfy We will note the identity matrix in two different ways, depending on where the indices sit. On the one hand we have 1 D ≡ δ µ ν = δ ν µ = diag{1, . . . , 1}, and on the other we will also consider the Kronecker deltas δ = δ µν and δ −1 = δ µν . Identical notation wil be used for dimensions other than D.
As it is well known, any element of the group can be decomposed as successive products of the following transformations: • B-shifts where Ξ µν = −Ξ νµ .
• Factorized dualities Any Ψ ∈ O(D, D) can be created through succesive products of these elements. The following two transformations will be of special interest: • Full factorized duality: This transformation is obtained by applying factorized dualities over all directions • β-shifts: where β µν = −β νµ , and as can be seen is a product of a full factorized T-duality, a B-shift and another full factorized transformation. For this reason it is also named TsT transformation.
As explained in (2.5), the O(D, D) group acts linearly on the generalized frame We can then analyze how O(D, D) transformations act on D-dimensional fields Using the O(D, D) identities (3.2) it can be shown that both expressions for Q ′ are equivalent. As explained before, in order to match degrees of freedom with supergravity we have to gauge fix and so we can work with a single vielbein e µ This is inconsistent with the duality rules (3.9) because both vielbeins transform differently. For this reason, the gauge fixing spoils the O(D, D) covariance of the generalized frame Namely, this matrix is not of the form (3.12). Another way to see this is noting in (3.9) that T (e (−) ) and T (e (+) ) are different. We then have to bring it to the form (3.12) through a compensating double Lorentz transformation where the sub-label c indicates that the Lorentz transformation is not generic, but a specific com- 15) and the second rewriting follows by using the identity Any choice for e ′ is purely conventional because they are all related by Lorentz transformations. From now on we will choose e ′ = M −t e . From (3.14) we can extract G ′ and B ′ as the symmetric and anti-symmetric part of Q ′ . Notably, using the O(D, D) identities, the transformations can be rewritten in a democratic way by defining shifted fields G * and B * The result for the O(D, D) transformations of the vielbein and two-form is: (3.20) The two ways of writing G ′ are equivalent due to (3.16). This is not surprising because both transformations correspond to the two ways of selecting e ′ discussed above (3.18) and in most cases this symmetry must also be gauged for consistency. Taking into account the parameterization of the generalized dilaton (2.14) and that the transformation of the determinant of the metric is given by Det( We would like to emphasize at this point that the results of this sections trivialy extend to the case of local O(D, D) × R + transformations. The action and equations of motion will not in general be invariant under such transformations, but the fields would transform in precisely the same way as explained here. We also stress that after the transformation the dual space is coordinatized by new coordinates X ′ , so the effect of the transformation is not only to rotate the fields, but also to change their coordinate dependence. In the case of rigid transformations both set of coordinates are related by (2.5), but in more general cases the relation is less clear. For concreteness, let us briefly discuss the coordinate dependence of the fields The original background E(X) is rotated with Ψ(X, X ′ ) in such a way that the product Ψ(X, X ′ )E(X) depends only on X ′ . Then, the compensating double Lorentz transformation depends on the dual coordinates O c (X ′ ), and we end with the dual background depending on the dual coordinates E ′ (X ′ ).
On the RHS it looks like there is some dependence on the original set of coordinates, but in reality there is not, the entire RHS is a function of X ′ only. For this reason, we are allowed replace on the RHS X → X ′ at no cost and avoid keeping track on the distinction between coordinates. Still, for clarity we will keep the distinction throughout the paper. Now we briefly move to the transformation of the generalized fluxes (2.12) and its impact on the torsionfull spin connections, as this will be useful later. Prior to the gauge fixing the generalized fluxes are O(D, D) invariant, but as we saw, the gauge fixing requires a compensating Lorentz transformation (3.14) under which the three-form fluxes transform where the antisymmetrization only affects indices [ABC]. We can extract from here the transformation of the components of the fluxes (2.18), namely the torsionfull spin connections This can also be obtained by direct computation from (3.20).
We now discuss the embedding of O(d, d) into O(D, D). To this end, the external components remain unchanged under the action of the duality group which only affects the internal space, namely with a, b, c, d being d × d matrices. These internal matrices can be rewritten in terms of an O(d, d) object which is the internal version of Ψ. It will be always possible to get Ψ from ψ using the trivial embedding (3.26).
Introducing this into (3.20) and decomposing the D-dimensional fields we can get a component version of the transformations where we defined the internal version of M where Q d is the matrix notation for the internal d × d components of Q, namely Q mn . To complete the picture, we notice that Det(M) = Det(M) and so Because it is going to be relevant later on, we present here the reduced form of O c in (3.15) given by where O c is exactly the internal version of O c with the internal vielbeins e d = e m a and the internal flat metric g d = g ab .

Generalized Dualities
In this section we give a brief review of generalized T-dualities and their embedding into O(D, D) × R + . Before moving to a case by case study, we first introduce a common starting point to set the notation.
Consider a group G acting freely on a manifold M . This means that given g ∈ G and p ∈ M , if g · p = p then g = e is the identity element. This permits to take a set of adapted coordinates on the target space (X, g) where g ∈ G, and X m are the spectator fields (or external coordinates) that label the orbits of G. As we explained before, and will discuss largely in this section, generalized dualities are represented by certain local O(D, D) × R + transformations that act exclusively on the twists that contain the information of the internal background. These are independent of the external coordinates, which then play no role in identifying the O(D, D) × R + elements associated to the generalized dualities. They do however play a major role when it comes to computing higher derivative corrections (as discussed around (2.55)-(2.56)), but we will concentrate on that in the next section. When the expectator fields are frozen to a trivial value, the action of G on M becomes transitive, meaning that any two points p 1 , p 2 ∈ M are always connected trough some g ∈ G such that g · p 1 = p 2 . In this case all the orbits become isomorphic to the manifold M itself and so we have a group manifold M = G, such that g ∈ G are the points in M parameterized with coordinates Y m .
the free and transitive right-action of G on M is carried by left-invariant vector fields k i ∈ g that transform the coordinates as The effect on the group element g ′ = g + δg can be obtained in two different but equivalent ways: through the right action on g ′ = g e ǫ i ti or by a change in the coordinates (3.34). This gives the relations We now define the following quantities The first is the adjoint action of g defined by matrices a i j , and the last two are the left and right invariant one-forms, respectively. It is easy to see from (3.35) that the following relations hold which in turn imply that the left and right invariant one-forms satisfy the Maurer-Cartan equations where L is the Lie derivative and [ , ] the Lie bracket. The last identity follows from noticing that R i = a i j k j and ∂ m a i j = −L m k a i r f kr j .

Abelian T-duality
This is the simplest case of a generalized duality that relates backgrounds with Abelian isometries. When the original background posses d Abelian isometries, there is a set of commuting killing The Abelian algebra f ij k = 0 allows to choose adapted coordinates for which all fields are independent of Y m and the compactified sigma model takes the form where Q = G + B contains the different components of the metric and B-field which depend on X m only. As explained before, for our purposes we could very well freeze the expectator fields X m and restrict attention to the internal sector, but we will keep track of them for the moment. Here we are neglecting the dilaton coupling which is going to be treated separately. The transformation (3.40) is a symmetry of the sigma-model as long as the fields satisfy the isometry conditions The way Abelian T-dualities emerge as symmetries was discussed by Buscher [4]. Beginning with the Lagrangian (3.41) one follows a 3-step recipe: (1) Gauge the global isometries, and then pick a gauge in which Y m = 0 (3.43) (2) Demand that the gauge fields behave like pure gauge by adding Lagrange multipliers Y m (3) Integrate A ± out and end up with the dual theory in terms of the dual coordinates Y m and the dual background Q d It can also be shown that the transformation of the dilaton is given by [4] (see also [31]) where Q d is the matrix notation for the internal components of Q.
The duality transformations (3.45) can be compared with the general way in which an element ψ ∈ O(d, d) acts on the background fields (3.29). This requires matching internal indices in both expressions by introducing δ matrices to relate tildes with primes and so Just to remind the reader, both coordinates and fields with tildes and primes refer to the dual space.
The former carry an unconventional index structure due to the way they are obtained through the Buscher procedure. The latter are defined to coincide componentwise to the former, in such a way that the standard index structure is restored through Kronecker deltas.
In this case it is obvious that both set of gaugings belong to the same duality orbit, and then give rise to the same physics, namely that of an ungauged supergravity.

Non-Abelian T-duality
The non-Abelian counterpart of T-duality [5] now relies on the target space possessing d noncommuting isometries L k Q = 0, generated by a non-Abelian group G with killing vectors satisfying To facilitate contact with the discussion at the beginning of this section, this is the first equation in (3.39). The sigma-model is where now the internal dependency is encoded in the right-invariant one-forms R m i , which act as vielbeins exchanging algebraic i, j = 1, . . . , d and curved m, n = 1, . . . , d indices This way of writing the background shows that the whole dependence on the internal space is encoded in the Maurer-Cartan forms R(Y ) m i , while the components Q(X) ij depend only on the spectator fields (external coordinates). Equation (3.52) is useful to note the difference with the Abelian case (3.41), where the Maurer-Cartan forms were trivial R m i = δ m i and so using algebraic or curved indices was equivalent.
There is a Buscher-like procedure built by De la Ossa and Quevedo [5] that leads to an equivalent dual background. The procedure closely follows the one performed before with the difference that the auxiliary fields A ± and their strength-energy tensors F +− are now valued in a non-Abelian algebra. The result is given by Regarding the dilaton field, its transformation can be obtained from [32] with a defined in (3.36), and this reduces to the standard form found by Quevedo and de la Ossa [5] when the algebra is uni-modular f ij j = 0. To cast this transformation in an O(d, d) format, we first need to express everything in terms of curved indices and then change from the tilde convention to the prime convention as we did in (3.47). To curve the indices we use the Maurer-Cartan forms of each space. For the original background we rotate with R m i , which connects Q mn = R m i Q ij R n j , and for brevity we will keep noting this with matrix notation Q d even though now this is a curved object. The dual background happens to carry an Abelian algebra and so algebraic and curved indices are indistinguishable and related by δ m i . For instance in the internal sector we have Finally, we lower the indices with δ mn as we did in (3.47) and for brevity we introduce a new mixedindex Kronecker's delta δ mn δ i n ≡ δ mi ≡ δ. After this procedure, (3.53) leads to while for the dilaton we have As discussed in (3.23), the RHS of these equations look like there is a dependence on the original set of coordinates through R(Y ), but after some work these equations can be taken to the form where Q d ≡ Q(X) ij and we are considering a non-trivial background for the dilaton Φ(X, Y ) = Φ(X) − 1 2 ln Det(a(Y )), which is isometric except in the non-unimodular case L ki Φ = 1 2 f ij j . It is then clear that the dual fields depend only on the dual coordinates Y ′ only.
The expressions (3.56) and (3.57) can now be compared directly with (3.29) and (3.31) to recognize the O(d, d) × R + transformation that connects the dual backgrounds Let us briefly discuss what happened above in the language of Gauged DFT. We started with the original generalized background corresponding to a geometric background with vielbein R m i , dilaton background − 1 2 ln Det(a) and vanishing 2-form flux. As such, the only components of the gaugings F IJK and F I are given by metric fluxes (2.33) These identities follow from the Lie bracket of R −1 (3.39) and the Jacobi identity. After the dualization, we ended with a different generalized background that yields the exact same gaugings (2.34) except for the vectorial flux F I which picks up a contribution from the trace of the structure constants In can be checked that (3.61) and (3.63) satisfy the consistency conditions (2.29).
We now discuss two distinct cases. If the group were unimodular f ij j = 0, then both set of gaugings (3.61) and (3.63) would coincide exactly. As a consequence, the Gauged DFT would remain invariant under the local O(d, d) × R + transformation (3.59) yielding the physical equivalence of both backgrounds, at least at the classical level. It would have been enough that both gaugings fell into the same orbit, but interestingly in this case they happen to coincide. Instead, if the group is not unimodular f ij j = 0, then both set of gaugings (3.61) and (3.63) fall into different duality orbits, and we loose guaranty that if the original background is a solution to the DFT equations of motion, so is the dual background. Note however that if the dual background ( the gaugings of this deformed background coincide with those of the original background Then, this background is indeed a solution to the equations of motion of DFT. We can interpret this fact as follows. The deformed background is a composition of two successive reductions: one with twist (1, λ ′ ) and another one with twist (U ′ , λ ′ ). The first twist (1, λ ′ ) produces a first gauging of DFT with fluxes The second twist reduces this Gauged DFT into another one with gaugings (3.65), which now happily fall into the same duality orbit than (3.61). Then, the local O(d, d) × R + (3.59) maps a solution (3.60) of ungauged DFT, to a solution (3.62) of a Gauged DFT with gaugings (3.66). Interestingly, the gauging (3.66) leads to the deformations of the DFT equation of motions, which on section happen to correspond to the so-called generalized supergravity equations [14], as discussed in [33], [34], [35].
The question remains on how to interpret the dual background (3.62) in the context of Gauged DFT. It is difficult to read a background from a generalized twist due to the double Lorentz symmetry. To avoid this ambiguity, it is instructive to build the generalized metric for the dual background where we set the scalar fluctuations to zero H IJ = δ IJ . This form of the parameterization in terms of a bi-vector is typical of globally non geometric backgrounds (see for example [26], [36]). It is clear from here that the background is locally geometric, but globally it corresponds to the wired case of a non-geometric background with a generalized paralellization that renders the gaugings purely geometric. Let us explain this a little further. The background is usually simple to read from the background generalized metric, while reading it from the twist matrix is cumbersome due to the redundancy produced by the choice of the internal double Lorentz gauge. This choice fixes the generalized paralellization [37]. It doesn't affect the background, but it does change the fluxes and then has a crucial impact on the lower dimensional physics. The paradigmatic case is that of a torus parallelized in a funny way that yields the fluxes of a sphere [13] (see also [38]).

Yang-Baxter deformations
Yang-Baxter deformations [39] relate backgrounds associated to integrable systems [40]. They are based on an R ij -matrix (not to be confused with the right-invariant one-form R m i ) satisfying the algebraic equation where c ∈ [−1, 0, 1], X, Y ∈ g and [ , ] is the Lie-bracket of the isometry algebra of the background to be deformed. The case c = 0 corresponds to classical YB equations (CYBE) (also called homogeneous equations) and c = 0 leads to so-called modified classical YB equations. The latter cases leads to inhomogeneous YB deformations, sometimes called η-deformations, and they have been widely study in the context of AdS 5 × S 5 backgrounds [41]. Here we will concentrate on CYBE only, which lead to homogeneous YB deformations, because its connection to NATD is simpler. These transformations preserve conformal invariance if the R-matrix is unimodular [42] R ij f ij k = 0 . It was conjectured in [43] that the homogeneous Yang-Baxter model can be obtained by applying NATD to the original background, with respect to an isometry group determined by the R-matrix. This conjecture was proven in [44] and [45] for principal chiral models where rules were established for connecting NATD and YB models.
Picturing YB deformations as NATDs requires a dressed R operator in the NATD background (3.53), where η is called the deformation parameter.
In [46] the NATD transformations for the Green-Schwarz (GS) superstring with a generic isometry group were derived. Using the rules between NATD and YB, the authors also deduced the form of homogeneous YB deformations for a generic GS sigma model given by is nothing but the curved version of the dressed R operator. Using the killing equations and closure of the algebra, the CYBE (3.68) translates into It can be shown that the isometric condition for the background fields and the uni-modularity condition ensure that both F IJK and F I remain invariant [16], [30], [52]. This can be seen by splitting where L ki is the generalized Lie derivative. If the twist U is generalized isometric with respect to the generalized killing vector k i , then the first term in (3.77) vanishes, while the second vanishes due to the YB equation (3.74). Then, the fluxes F IJK remain invariant. Regarding the fluxes F I , the first term in (3.78) vanishes if the twist λ is generalized isometric, while the second term vanishes if the group is unimodular (3.69). If this is the case, then the dual gaugings fall into the same duality orbit. If not, a procedure similar to (3.64) is required in order to interpret the dual background as a solution to a deformed theory. Note however that in this case, the dual vectorial fluxes would be non-constant, and so it is unclear to us if they can be generated through a twist in Gauged DFT. A similar discussion on this point will take place in PL T-duality.

Poisson-Lie T-duality
In [6], Klimcik and Severa brilliantly abandoned the requirement of isometries as the guiding principle for duality, replacing it by a higher algebraic structure that relates dual models, in which isometries only show up in special cases. We will review the procedure restricting attention to the internal sector, so the expectator fields will be frozen. The starting point is then the internal sector of a generic sigma-model where the group G acts freely and transitively. It transforms the coordinates as in (3.34) δY m = ǫ i (σ ± )k i m , inducing the following change in the action where we defined the Noether currents Neglecting the global term in (3.80), the Abelian and non-Abelian T-duality scenarios are recovered by considering G as the isometry group of the target space in which k are the killing vectors. The interesting point is that the invariance of the action can still be satisfied without isometries. The idea is to think of J i as the components of an element J of a dual algebra g ′ with an associated Maurer-Cartan equation The invariance of the action, namely the vanishing of (3.80), leads to a non-isometric condition on the background

84)
and analyzing the closure of the algebra over it leads to a bi-algebraic condition [53], [54] f ij s f ′kr which can be conceived as the mixed components of the Jacobi identities of an extended algebra To enforce that both algebras appear on an equal footing in this framework, a dual background Q ′ mn is introduced together with a dual version of the algebraic identities (3.36)-(3.39) Combining the bi-algebraic condition (3.85) with the introduction of a non-degenerate, ad-invariant bilinear form , satisfying one can identify g and g ′ with the maximally isotropic subalgebras of a Drinfeld double D [53]. It was shown in [6] and [55] that the sigma-models associated to Q and Q ′ are related by a canonical transformation, so both backgrounds satisfy the same equations of motion.
Using the structure of Drinfeld doubles one can build solutions to the PL conditions (3.84) and (3.87) given by [6] and the matrices a(g), c(g), a ′ (g ′ ) and c ′ (g ′ ) are defined by the adjoint action while for the dual matrices we have The expression for the dilatons were originally given in [56] and latter improved in [32] in the context of PL-plurality (see also [17]) where, Φ can be taken to be a constant, that on general grounds would depend only on the expectator fields 5 . 5 PL-duality works even if Φ depends on the internal coordinates [32].
Elimination of Q and Φ in (3.90) and (3.94) leads to Notice once again that although here it looks like the RHS depends on the original set of coordinates Y through L(Y ), R(Y ) and π(Y ), in reality they only depend on Y ′ as is clear from (3.90) and (3.94). These expressions (3.95) can now be compared with (3.29) and (3.31) to recognize the O(d, d) × R + transformation that connects the dual backgrounds and those of the dual background by Before we compute the gaugings, let us show how the previously introduced expressions can be cast into a double language (see for example [33]). Grouping the generators into a double generator T I = (t i , t ′i ) permits to cast the maximal isotropic condition (3.89) in terms of the O(d, d) invariant matrix and also regroup the algebra (3.86) in an O(d, d) covariant fashion The ad-invariant condition over , can then be written as (3.101) Of course we will see that these generalized structure constants F IJ K are exactly the gaugings generated by both backgrounds. We finally point out that the adjoint actions (3.92) and (3.93) can be also combined into an O(d, d) form where we read Ad g −1 and Ad g ′ −1 from (3.92) and (3.93) and then inverted the matrices. These matrices can be contracted with double left-invariant 1-forms in order to obtain the twist matrices (3.97) and (3.98) Having written everything in double language, it is now obvious that we can rotate every object carrying indices I, J, K, . . . with rigid elements h ∈ O(d, d), which is simply a renaming that does not change the results. In the language of Gauged DFT this simply amounts to translations withing a fixed duality orbit, as discussed around (2.35). In the context of generalized dualities, these rotations are known as PL T-pluralities [32]. This is a generalization of PL T-duality which considers that a Drinfeld double D, can be decomposed in several maximally isotropic subalgebras g and g ′ . Together with the Lie algebra of the Drinfeld d, every such decomposition (d, g, g ′ ) is known as a Manin triple M(D). An important remark is that for any D at least we have two Manin triples (d, g, g ′ ) and (d, g ′ , g), connected by a full factorized O(d, d) rotation, which from the point of view of the bialgebra are distinct objects. Any such decomposition will give rise to a different background but all of them will be dual to each other. In this scenario, all models are connected by rigid O(d, d) rotations preserving the bi-algebra (3.86) and the maximally isotropic condition (3.99) We can finally compute the gaugings in the context of Gauged DFT defined by the twists (3.97) and (3.98), yielding 6 where the structure constants of the bi-algebra (3.86) turn out to be the non-vanishing components of the generalized fluxes, as expected. Keeping track of the origin of the fluxes, it can be seen that in the unprimed background the geometric-type fluxes come from R-vielbein metric fluxes, while π introduces the non-geometric Q-type flux given by the structure constants f ′ of the dual algebra. Curiously, in the primed background the Q-type fluxes are generated by R ′ , and the geometric ones come from the bi-vector π ′ (this a generalization of the NATD case where we saw that the dual background consisted of a globally non-geometric space (3.67) with a generalized parallelization that rendered the fluxes geometric).
As in the NATD case, we have two different situations. If the groups are unimodular f ij j = f ′ ij j = 0, then the original and dual gaugings fall into the same orbit, both backgrounds are solutions to 6 To facilitate the computation of the fluxes, we list some useful identities (see also the appendix of [55]). The ad-invariance condition of the bilinear form (3.101) implies Analogous identities can be obtained for the dual objects by just adding/removing primes and exchanging the position of all indices. We finally point out that he derivatives of π and π ′ can be obtained by deriving the adjoint actions (3.92) and (3.93). Also (3.39) must be used.
ungauged DFT, and we are done. If not, the original and dual gaugings (3.108) happen to fall into different orbits due to the discrepancy between the vectorial components. Moreover, these gaugings are not constant, as they carry a dependency on the internal coordinates through the adjoint matrices. Interestingly, they still happen to satisfy the consistency constraints (2.29). The action and equations of motion of DFT depend on the gaugings through the generalized fluxes (2.24). Then, the discrepancy between gaugings (3.108) can be cured by deforming the original and dual DFT through shifts in F A intended to annihilate F I and F ′ I respectively. While in the case of NATD these shifts were produced through a gauging procedure (3.64), it is unclear to us if similar steps can be taken in this case. The required deformations again fall into the category of the so-called generalized supergravities [14], as shown in [57], [17]. So again, as in the NATD case with nonunimodular gaugings, the local O(d, d) × R + transformation connects solutions of deformed DFTs.
As mentioned above, Poisson-Lie T-duality is as a generalization of Abelian and non-Abelian Tdualities and so these results must contain both of them as particular cases. Lets see how this works. To do this, we need the explicit infinitesimal expressions for π and π ′ which can be obtained using the exponential maps for g = exp(Y i t i ) and g ′ = exp(Y ′ i t ′i ) in the definition of the adjoint actions For the other objects, namely L, R and a and their duals, it will be enough to know that they are trivial for Abelian algebras Then, for Abelian T-duality we have f = f ′ = 0 and so Inserting this into (3.95) we obtain Likewise, for non-Abelian T-duality (3.53) we have f ′ = 0 but f = 0 so Also, since the dual algebra is abelian R ′ = L ′ = δ, a ′ = 1. Inserting this particular case in (3.95), we get

Generalized dualities and higher derivatives
In this section we finally arrive at the main result of this paper: a general formula for first order higher derivative corrections to generalized dualities. We begin by refreshing the relation between O(D, D), gauge fixing and the need for compensating double Lorentz transformations.
As explained in Section 2.3, higher derivatives deform the double Lorentz transformations of generalized fields, and consequently of their components. For this reason, the components are not the usual fields in supergravity, but instead are related to them through field redefinitions. In order to distinguish them we use the notation that the components of generalized fields carry an overline ē (±) ,B andΦ. These fields transform as follows with respect to generic (possibly local) O(D, D) × R + transformations (3.9) On the other hand, we found the full action of finite double Lorentz transformations to first order in α ′ on these components (2.53) whereŌ (±) are first order corrected elements of the different factors of the double Lorentz groups, that to lowest order are given by O (±) , and We explained in Section 3.1 that the gauge fixingē (−) =ē (+) =ē is inconsistent with the action of T = O(D, D) × R + (4.1), unless the gauge is restored through a specific compensating Lorentz transformation L c . One can then define the combined transformation on the components as and choose L c in such a way thatē (−)′ =ē (+)′ =ē ′ is achieved. For concreteness let us write these transformations explicitlyē where (4.7) As explained, in order not to have ambiguities in the definition ofē ′ we needē (+)′ =ē (−)′ . One way to achieve this is by choosing the following compensating double Lorentz transformation whereŌ c contains the compensating Lorentz transformation that we made at zeroth order (3.15) now promoted to over-lined fieldsŌ After this choice, we haveē which corresponds to the first order correction to (3.18).
In this gauge, whereŌ Finally, noticing that M −t e is what we called e ′ , we use the equality ω (−) (e ′ ) = ω (−)′ together with (3.25) to arrive at (4.14) We have then finally arrived at the first order in α ′ generalized T-duality transformations in the DFT scheme: where (4.16) These are the first order corrections to the equations (3.20) and (3.22). They capture any generalized duality, encoded here in generic local O(D, D) × R + transformations, for any choice of the parameters a and b that control the first-order corrections in the deformed DFT. These expressions are valid in the DFT scheme, namely for the components of the duality covariant fields after the gauge fixing. These are not the fields that appear in supergravity, but are related to them through field redefinitions, as we discuss in the following section.
The gauge fixingē (+) =ē (−) =ē requires that L(e (+) ) = L(e (−) ) in (4.3). This forces a relation between that is solved as followsŌ It is clear that these are elements of the Lorentz group for any value of the parameter γ because the matrix A is antisymmetric. Note that while to lowest order the two elements are forced to coincide (this is the usual case in which the double Lorentz symmetry breaks to its diagonal subgroup), higher orders make the two transformations differ. Since γ can be chosen at will, we set its value to γ = 1. This implies the following Lorentz transformations for the gauge fixed fields where Σ is given in We explained at the end of Section 2.3 why, even at higher orders, the local O(D, D) × R + transformations map solutions into solutions of DFT. In the context of Gauged DFT this is realized rather trivially: the transformation keeps the gaugings into the same orbit and then works as a symmetry of the Gauged DFT. Even if the orbits are different one can make sense of the transformation as a solution generating technique between deformed theories, as we discussed for instance when gaugings are non-unimodular. It is then natural to ask why this extreme simplicity is no longer reflected in the results of this section. The reason is that here we have introduced a necessary gauge fixing to make contact with supergravity (in the DFT scheme). This gauge fixing breaks the O(D, D) covariance, which has to be restored by compensating Lorentz transformations that induce corrections to the generalized duality transformations of the gauged fixed fields.

Supergravity schemes
The overline on fields in the previous section indicates that they are components of the generalized fields in DFT, and so we call this set of fields the DFT scheme. In this scheme the frame field receives a first order Lorentz transformation inherited from the generalized Green-Schwarz transformation (2.40), and so it is not the frame field in supergravity. However, it is related to it through a first order Lorentz non-covariant field redefinition. The same is true for the dilaton and two-form (although in some cases the Lorentz transformation of the two-form cannot be redefined away). So the fields in the DFT scheme (with an overline) and the fields in supergravity (without an overline) are related bȳ e = e + ∆e ,B = B + ∆B ,Φ = Φ + ∆Φ . (4.20) The correction ∆ depends on the supergravity scheme to be considered, and is defined up to covariant Lorentz redefinitions. The non-covariant part is fixed by The only case in which the two-form can be taken to be a Lorentz invariant field L(B) = B is when a = b, which corresponds to the bosonic string [21]. Otherwise it carries a Green-Schwarz transformation. Different supergravity schemes [58]- [59] correspond to different choices of (∆e, ∆B, ∆Φ) related by Lorentz covariant field redefinitions.
Applying a generalized duality toē ′ leads tō We know from (4.15) whatē ′ is in terms ofē, and from (4.20) whatē is in terms of e, so we can readily computē were indicates that we used the identity (A + ǫ) −1 A −1 − A −1 ǫA −1 for small perturbations, and truncated the result to first order in α ′ . We then have where (∆e) ′ is the result of performing a zeroth order O(D, D) transformation on the fields that define the redefinition ∆e.
For the metric the above results imply This expression can be improved by using the following O(D, D) identities such that the first, second and third terms of G (1) are combined into a single symmetric term For the two-form we follow the same procedure (4.27) After introducing B * as in (3.19) and using exhaustively the O(D, D) identities, we can get a similar result as for the metric Both results (4.26) and (4.28) can then be merged into a single expression in terms of Q ′ = Q (0) +Q (1) . The final result for first order corrections to generalized dualities is given by: (4.29) We have then extended the result of the previous subsection to be applicable to generic schemes related by field redefinitions from the DFT scheme. This reduces to (4.15) when ∆e = ∆B = ∆Φ = 0.
Finally, we can split the result into external and internal components to obtain the corrections to (3.29), as this might be useful for applications to specific solutions 30) with (here O is the internal submatrix of O c , we ommit the sub-label in O c to lighten the notation) (4.31) The fields without an overline must transform covariantly under Lorentz transformations (4.21). We can then separate ∆ into a non-covariant part, and a covariant part. The former is unambiguously defined, and the scheme in which ∆ contains only the non-covariant part was named the Bergshoeffde Roo (BdR) scheme in [21], after [59] ∆G (BdR) with ω (±)2 = ω (±) µα β ω (±) νβ α . As explained, it is not always possible to make the two-form Lorentz invariant, and interestingly in the BdR scheme the two-form coincides with the two-form in the DFT scheme. Expressing (4.29) in this particular scheme gives    (4.37) To move forward, we need to specify a particular scheme. Considering the Bergshoeff-de Roo scheme, we can use (4.33) which after some work can be rewritten as We consider the simple case of a single internal isometric direction, and then perform a splitting as we did in Section 3.1.2 for the zeroth order. In this case the results are where we defined Ω ≡ 1 2 ω (+)2 . These results are the same as the ones obtained in [60] for the heterotic string after identifying B and setting the α parameter in that paper to 1 2 (see eqs. (39,42,70,74,75,76)).

Yang-Baxter
We now move to a different generalized duality for backgrounds with non-Abelian isometries. In [61] it was shown that after applying unimodular homogeneous YB transformations over bosonic string solutions at order α ′ , the resulting background could be corrected to satisfy the equations of motion. This was done for backgrounds with vanishing NSNS fluxes and up to second order in the deformation parameter η. Soon after, in [30] it was realized that the same result could be obtained by considering these particular generalized dualities in the context of DFT to order α ′ . In this case the deformed background was obtained at all orders in η and the original background was allowed to have NSNS-fluxes. As expected, the result reduced to the previous one after setting the particular conditions of [61].
Our general formula for higher-derivative corrections to generalized dualities includes this scenario as a particular case. We will show here that the results of [30] are recovered, a task that will turn out easy because we are using a notation similar to the one used there. To see this, we first notice that (3.72) can be trivially extended to D-dimensions by where k i µ are extended by introducing the identity map on the external directions. The same can be done for the Maurer-Cartan form and so the expression (3.72) can be brought to Q ′ = Q (ηΘQ + 1) , Φ ′ = Φ − 1 2 ln (Det(ηΘQ + 1)) .  [30]. Finally, for the dilaton field instead of using our general formula, the more straightforward way to match results is noticing that in the scheme (4.42) one has e −2d = e −2Φ Ḡ = e −2Φ √ G 1 + 1 24 which is exactly the expression given there in eq. (4.16).

Outlook
A number of questions arise: • Explicit solutions. It would be interesting to apply our result to specific examples. Higherderivative corrections to Abelian T-duality have been applied in different contexts, such as corrections to entropy and black-hole solutions [62], [63] and cosmological backgrounds [64]. Higher-derivative corrections to Yang-Baxter deformations were recently considered in [30]. There have also been some analysis on higher-derivative corrections to non-Abelian and PL dualities [65].
• Classification of generalized dualities. An interesting observation is that the framework of Gauged DFT allows to envision further extensions of generalized dualities, beyond those discussed here. In particular it might offer a classification through classifications of duality orbits on the one hand, and on the other through the characterization of the degeneracy in the space of duality twists that fall into the same orbit. Steps in this direction were given in [13] and [66]. There are a priori no obstructions in finding examples of generalized dualities in Gauged DFT that go beyond PL T-plurality. An interesting case of study is the so called E-models [67] recently discussed in the context of DFT in [33].
• Extensions to higher orders. The whole construction in the paper was based on the first order generalized Green-Schwarz transformation (2.40) introduced in [21]. In order to proceed to even higher orders, we need further corrections to the generalized Green-Schwarz transformation. Interestingly, for the heterotic string these corrections are known non-perturbatively (through the so-called generalized Bergshoeff-de Roo identification), and perturbatively to second order in α ′ [68]. Soon, an all-order proposal to corrections in the general bi-parametric case will appear [69], where the second-order corrections will be worked out explicitly. The strategy applied here, together with these results will permit to extend our computations to second order in α ′ .
• Exceptional Drinfeld Doubles and maximal supergravity. The results in this paper are at most compatible with half-maximal supergravity. Generalized dualities in the context of maximal supergravities gained renewed interest after the proposal for non-Abelian dualities of RR fields [70]. Type II and M-theory give rise to rigid U-duality transformations upon compactifications on tori. Interestingly, the idea of generalized U-dualities was recently introduced in [71] and further discussed in [72]. Looking for higher order corrections to generalized U-dualities is out of reach at the moment, because these corrections are not even known in the Abelian case.
There are promising steps in this direction [73], systematics in the writing and counting of interactions is crucial [74] because higher derivatives appear in maximal supergravity at order α ′3 , and so even the simplest corrections are hard to handle. Still, there is at the moment no higher derivative formulation of Exceptional Field Theory [75] nor Type II DFT [76] (for a review see [77]), but generalized Scherk-Schwarz reductions have been extensively investigated [78] and surely constitute the proper framework to deal with generalized U-duality, in the same sense that Gauged DFT is the proper framework to deal with generalized T-duality.
We hope to make progress in these and other directions in the future.