Non-Abelian T-duality as a Transformation in Double Field Theory

Non-Abelian T-duality (NATD) is a solution generating transformation for supergravity backgrounds with non-Abelian isometries. We show that NATD can be described as a coordinate dependent O(d,d) transformation, where the dependence on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. Besides making calculations significantly easier, this approach gives a natural embedding of NATD in Double Field Theory (DFT), which provides an O(d,d) covariant formulation for effective string actions. As a result of this embedding, it becomes easy to prove that the NATD transformed backgrounds solve supergravity equations, when the isometry algebra is unimodular. If the isometry algebra is non-unimodular, the generalised dilaton field is forced to have a linear dependence on the dual coordinates, and hence the resulting background solves generalised supergravity equations.


Introduction
Non-Abelian T-duality (NATD) is a generalization of T-duality for strings on backgrounds with non-Abelian isometries [1]- [5]. Although it is not as well established as T-duality is as a string duality symmetry, it works well as a solution generating transformation for supergravity. The rules for the transformation of the fields in the NS-NS sector, namely the metric, the B-field and the dilaton field has been known for a long time. Recently, NATD has gained a new interest, as the rules for the transformation of the fields in the RR sector of Type II strings has also been found [6]. This has been applied to many supergravity backgrounds by various groups, especially to backgrounds that are relevant for AdS-CFT correspondence, for example see [7]- [16].
Recently, a compact formula for the transformation of the supergravity fields for a generic Green-Schwarz string with isometry G has been obtained by [17], where they also showed that the sigma model after NATD has kappa symmetry. This then means that the resulting supergravity background is a solution of the generalised supergravity equations (GSE), which have recently been introduced in [18] as a generalisation of supergravity equations, see also [19].
To be more precise, when the isometry group G is unimodular, the dualised sigma model is Weyl invariant and the target space is a solution of standard Type II supergravity equations. If G is non-unimodular so that the structure constants of the Lie algebra of G is not traceless, the trace components give rise to a deformation of the equations to be satisfied by the target space fields to GSE.
One purpose of this paper is to describe the NATD transformation rules obtained in [17] as a coordinate dependent O(10, 10) transformation . For Abelian T-duality, the transformation rules for the supergravity fields in the NS-NS sector can be neatly described through the action of a constant O(d, d) matrix [20]. The RR fields are then packaged in a differential form, which can be a regarded as a spinor field that transforms under Spin(d, d). If the fields in the NS-NS sector transform under T ∈ O(d, d), then the spinor field that encodes the RR fields transform under S T ∈ Spin(d, d), which is the element that projects onto T under the double covering homomorphism ρ between O(d, d) and Spin(d, d), that is, ρ(S) = T [21]. In a similar fashion, we show in this paper that the NATD transformation of the supergravity fields in the NS-NS sector can be described through the action of an O(d, d) matrix, this time not constant but with an explicit dependence on the coordinates of the dual theory. The dependence on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. The transformation of the RR fields is then automatically determined by the corresponding Spin(d, d) matrix, as in Abelian T-duality. We would like to note that we had already presented the NATD matrix we give in this paper at a workshop at APCTP, Pohang [24]. Very recently, a paper has appeared which also views NATD as an O(d, d) transformation [22]. See also [23], which has a similar approach to NATD.
Besides making calculations significantly easier, our approach gives a natural embedding of NATD in Double Field Theory (DFT), which provides an O(d,d) covariant formulation for effective string actions [25]- [35] by introducing dual, winding type coordinates. When the so called section constraint is satisfied, DFT reduces to standard supergravity, or an O(d, d) transformation thereof. As a result of this embedding, we manage to give a simple proof of the fact that the transformed backgrounds solve supergravity equations, when the isometry algebra is unimodular. If the isometry algebra is non-unimodular, we show that the generalised dilaton field of DFT is forced to have a linear dependence on the dual coordinates, and that this implies the resulting background should solve GSE. Our approach in this paper is very different from that of [17], where they prove that the dualised background satisfies the generalised supergravity equations by checking the kappa symmetry of the transformed Green-Schwarz sigma model. Here, we consider the transformation of the fields within DFT and directly check that the dual fields satisfy the field equations of DFT in an appropriate frame, where they reduce to (generalised) supergravity equations. NATD (and the related Poisson-Lie T-duality) has been studied in the context of DFT also in the papers [36], [37], [38], and very recently in [22].
As the O(10, 10) matrix T that produces the NATD fields is not constant, it is not immediately clear that the transformed fields satisfy the field equations of DFT. For this reason, we find it useful to utilize the framework of Gauged Double Field Theory, which is obtained by a duality twisted (Scherk-Schwarz) reduction [39] of DFT [40]- [43]. GDFT is a deformation of DFT, determined by the fluxes associated with the twist matrices that define the duality twisted reduction anzats. Our strategy is as follows: We start with a solution of Type II supergravity. Since Type II supergravity can be embedded in DFT, one can construct corresponding DFT fields which constitute a solution for DFT in the supergravity frame, that is, in the frame in which the section constraint is satisfied trivially by demanding that the DFT fields have no dependence on the winding type dual coordinates. If the space-time metric has an isometry symmetry G, which is also respected by the B-field and the RR fluxes (not necessarily the gauge potentials), we can extract DFT fields out of the original ones, which satisfy the field equations of GDFT determined by the geometric flux associated with the isometry group G. We call these fields untwisted DFT fields. The DFT fields corresponding to the NATD background are obtained by acting on these untwisted fields with the O(10, 10) NATD matrix we present in this paper. As we will demonstrate in the body of the text, the fluxes associated with this NATD matrix is exactly the same as the geometric flux associated with the isometry group G. This makes it possible to prove that the DFT fields corresponding to the NATD background also satisfy the field equations of DFT, again in the supergravity frame, provided that G is unimodular. Equivalently, the corresponding supergravity fields satisfy the field equations of Type II supergravity. On the other hand, if G is not unimodular, we show that the generalised dilaton field is forced to have a linear dependence on the winding type coordinates, and we are not in the supergravity frame anymore. DFT in such a frame was studied in the papers [44] and [45], where it was shown that the equations of DFT reduce to GSE. As a result, the NATD fields indeed form a solution of GSE, when G is not unimodular. For related work where the relation between non-unimodularity of G and the generalised supergravity equations is explored, see [46].
The three key facts that we will prove and use in this paper are the following: i) The fluxes associated with the isometry group G and the NATD matrix T are exactly the same. ii) Duality twisted DFT fields, which we generically call φ(x, Y ) = U (Y ).φ(x) 1 satisfy the field equations of DFT if and only if the untwisted fields φ(x) satisfy the field equations of GDFT determined by the fluxes associated by the twist matrix U . iii) Field equations of GDFT are O(d, d)/Spin(d, d) covariant, provided we also allow fluxes transform as generalised tensors. Utilizing (ii) and (iii), we show that a set of fieldsφ(x, Z) =Ũ (Z).φ(x), where the twist matrix U (Z) produces the same fluxes as U (Y ), will satisfy the field equations of DFT if and only if the fields φ(x, Y ) satisfy the field equations DFT. Together with (i), this proves that the NATD fields form a solution of DFT and hence of GSE, when in the appropriate frame.
The structure of the paper is as follows: In the next section, we give a brief review of O(d, d) structure of Abelian T-duality, first in the NS-NS sector and then in RR sector in subsection (2.2). This enables us to identify the coordinate dependent O(d, d) matrix that generates the NATD background in section (3). Also in this section (in subsection (3.1)), we demonstrate how the NATD of the background AdS 3 × S 3 × T 4 studied in [6] can be obtained by the action of the NATD matrix we have identified. Then in section (4), we study the embedding of NATD in DFT. The three key facts that we listed in the previous paragraph are proved in this section. The distinction between the unimodular and non-unimodular case is also discussed here. We finish the paper with discussions and outlook in section (5).
Note Added: While we were about to finalize the writing of this manuscript, the paper [22] appeared on the arXiv, parts of which have a significant overlap with the work we present here.

The Action of O(d, d) on curved string backgrounds
In this section, we review how Abelian T-duality can be described as an O(d, d) transformation, first in the NS-NS sector and then in RR sector, for which the duality group should be lifted to Spin(d, d). We closely follow [20] in (2.1) and [21] in (2.2).

Transformation of the fields in the NS-NS sector
Let g and B be the metric and the Kalb-Ramond 2-form field that describes a D dimensional supergravity background, with d commuting isometries. The string living on this background exhibits an O(d, d, Z) T-duality symmetry. Accordingly, there is an O(d, d, R) action, which acts as a solution generating symmetry in the low energy limit. Let us define the D × D background matrix Q(g, B) = g + B. (2.1) Due to the presence of d commuting isometries, it is possible to choose adopted coordinates X I = (x a , x µ ), I = 1, · · · , D; a = 1, · · · , d such that the background matrix does not depend on the d coordinates x a . Let us decompose the background matrix with respect to such choice of coordinates as This can be embedded in O(D, D, R) as followŝ be the new background matrix obtained by the above action of O(D, D, R) on Q. Then, it is well known that the transformed metric and the transformed B-field obtained as define valid supergravity backgrounds. That is, the O(D, D, R) transformation defined above acts as a solution generating symmetry.
For completeness, let us write the final form of the transformed background matrix Q ′ : For the resulting background to be a valid supergravity slution, the dilaton field should also transform under O(D, D, R) in the following way:

Transformation of the fields in the RR sector
Let us now discuss how p-form fields in the RR sector of Type II supergravity theories transform under the action of O(D, D, R) described in the section above. For this discussion we closely follow [21], see also [47].
In the democratic formulation of Type II supergravities [48], the 0,2 and 4-form fields in Type IIB and 1 and 3 form fields in Type IIA are combined with their Hodge duals to form sections of the exterior bundle even T * M for the first case and of odd T * M for the latter, where M is the D-dimensional space-time manifold. It is well known that these bundles carry the chiral spinor representations of P in(D, D), which is the double covering group of O(D, D). The transformation of the RR fields under T-duality is determined by this action of P in(d, d) on the RR fields, viewed as a section of the exterior bundle. More precisely, if the T-duality transformation in the NS-NS sector is realized by the O(d, d) matrix T , then the P in(d, d) element acting on the spinor field that packages the modified RR gauge potentials is S, where ρ(S) = T . 2 Here, ρ is the double covering map (2.11) Then, if χ is the spinor field that packages the modified RR fields we have Let us now discuss the transformation of the field strength / ∂χ under P in(D, D). This is important since RR fluxes are defined as As was discussed in [21], the transformation (2.12) does not imply / ∂χ → S / ∂χ. However, when one doubles the space-time coordinates as in Double Field Theory (DFT), which we will discuss in more detail in section (4.1), / ∂χ also transforms as a vector under P in(D, D) as / ∂χ → S / ∂χ. In DFT, the usual space-time coordinates are doubled by introducing winding type coordinates ν I , which combine with the space-time coordinates to form an O(D, D) vector X M = (ν I , x I )that transforms as 14) The transformation of / ∂χ implies that

NATD as an O(d, d) transformation
The aim of this section is to show that NATD can also be described, just like Abelian T-duality, through the action of an O(d, d) transformation. First, we give a brief review of the NATD rules for a generic Green-Schwarz string sigma model with isometry group G, which have recently been obtained in [17]. Then, we identify the O(d, d) matrix whose action on the target space produces the NATD background. As we will see, this O(d, d) matrix has an explicit dependence on the coordinates of the dual background, and the dependence is determined by the structure constants of the Lie algebra of G.
Non-Abelian T-duality can be applied by using the standard tools of the Buscher method. For a generic nonlinear sigma model with isometry group G, one starts with gauging the symmetry group (or a subgroup of it) and introduces Lagrange multiplier terms which constrains the gauge field to be pure gauge. Integrating out the Lagrange multipliers, one obtains the original model. Integrating out the gauge field gives the NATD model, for which the Lagrange multiplier terms play the role of coordinates on the dual manifold.
The NATD of a generic Green-Schwarz string sigma model with isometry group G has recently been obtained in [17]. Here we present their results (for bosonic G) and show that the new backgrounds can also be obtained by applying a coordinate dependent O(d, d) transformation. The best way to present the rules for transformation is to introduce coordinates which makes the isometry symmetry manifest. With respect to such coordinates one can write Here, σ a , a = 1, · · · , dimG are the Maurer-Cartan 1-forms on the group manifold G. Then g −1 dg = σ a T a , where T a are generators of the Lie algebra G of G. Then, if we call θ i to be the coordinates on the group manifold we have g −1 dg = l a i T a dθ i . After applying NATD with respect to G, one ends up with a sigma model which corresponds to the following background 3 The metric and the B-field in the transformed background are where ν a are coordinates of the dual manifold, which results from the Lagrange multipliers in the Buscher method. Here, where C c ab are the structure constants of the Lie algebra G with respect to the basis T a , that is, [T a , T b ] = C c ab T c . The transformation for the dilaton field presented in [17] is Now, let us write the above transformation rules in the terminology of the previous section. We define the background matrix Q = g + B and Q ′ = g ′ + B ′ . Then the above rules become Comparing this with (2.8) and (2.9) one immediately sees that the new background has been obtained by the action of the fractional linear transformation with the following O(d, d) matrix T embedded in O (10,10) in the way presented in the section above: Let us also check that the transformation rule (3.12) for the dilaton field can be obtained through the action of T by comparing it with (2.10). It is a well known fact that the transformation (2.6) implies for g ′ the following [20]: When T is as in (3.17) this gives detg ′ detg = detN, (3.20) and the two expressions (2.10) and (3.12) indeed match.
It is important to note that the dimension d of the isometry group determines whether the the T-duality matrix T acts within Type IIA/Type IIB or it involves a reflection which implies that a Type IIA solution is mapped to a Type IIB solution or vice versa. The former situation arises when d is even and the latter occurs when d is odd.
Since we have identified the O(10, 10) matrix that generates the NATD background via fractional linear transformation of the original background, we can immediately determine the transformed RR sector, as well. All we have to do is to find the P in(10, 10) matrix that acts on the spinor field that packages the modified p-form gauge potentials in the democratic formulation. The P in(10, 10) element S T that projects to the O(10, 10) element (3.17) under the double covering homomorphism ρ : P in(d, d) → O(d, d) can be found easily: where C is the charge conjugation matrix. For more details, see [35] and [43]. The factors S θ and S β in S T are the Spin + (d, d) elements that projects onto the SO + (d, d) matrix that generates the B-transformations and β-shifts with θ ab = ν c C c ab and β ab = ν c C c ab , respectively. Then the transformation of the p-form fluxes is An important remark is in order here. Recall from the discussion in section (2.2) that the transformation (2.15) is equivalent to the transformation (2.14). Also recall that the transformation (2.14) equivalent to the transformation (2.12), when S is constant. However, when S is not constant as in here, the two transformations (2.14) and (2.12) are not equivalent. Naively, one would have expected that the right transformation rule for the RR fields under NATD would follow from the transformation which would imply a different transformation rule for the field strength F that would also involve fluxes associated with S T (see section (4.3.2)). However, the right transformation rule is as in (3.22), as we will verify in the next section (3.1). Also in section (4.87), we will see that the transformed fields will constitute a solution of the GSE when the transformation for the RR fields is as in (3.22).

An Example:
Let us consider the simple example AdS 3 × S 3 × T 4 . This geometry arises as the near horizon limit of the D1-D5 system. The geometry has to be supported by 3-form Ramond-Ramond flux. We have Note that we also need the Hodge dual of the 3-form flux which is the following 7-form flux: Due to the presences of the 3-sphere in the geometry, one has a global SO(4) ≃ SU (2) × SU (2) isometry symmetry. It is possible to use one of these SU (2) groups to apply NATD. Writing the S 3 part of the metric as where σ a , a = 1, 2, 3 are the 3 Maurer-Cartan 1-forms for SU (2), we have and Now we apply the NATD matrix (3.17) on this background. Here, the structure constants that determine the NATD matrix are C c ab = ǫ c ab . This gives Writing this in spherical coordinates we have Now, let us look at the transformation of the RR sector. Similar to the Abelian case we form the differential form, which encodes the RR fluxes where we have decomposed a p−form RR flux G (p) according to how many legs it does have along the directions of the isometry group SU (2). The fluxes F (p−a) , a = 0, 1, 2, 3 have no dependence on the coordinates r, θ, φ. We map this differential form to a Clifford algebra element in the usual way. The difference we have here is that it is σ a and not dx a that we identify with the Clifford algebra element ψ a , for a = 1, 2, 3. On the other hand, for µ = 4, · · · , 10, dx µ is replaced with ψ µ , as usual. Here, ψ I are the Clifford algebra elements ψ I = 1/ √ 2Γ I , where Γ I are the Gamma matrices. For more detail, see [43]. For the example we consider in this section we only have 3-and 7-form fluxes, so the spinor field takes the following form: (3.22). First note that e −b F = F , since the B-field is zero on the original background. Let us first calculate S θ F = (1 + ν c ǫ c ab ψ a .ψ b ).F . As one can easily calculate, this gives Now we apply the charge conjugation operator [35] C on (3.40): Finally, we apply e b ′ = 1 − 1 1+r 2 ν c ǫ c ab ψ a .ψ b on (3.42): From F ′ we can read off the p-form fluxes of the dual background after now identifying ψ a with dν a . Since it is only the S 3 directions that have been dualized, we still have As a result we have and ⋆ is the Hodge dual with respect to the metric of the deformed background given in (4.29). These results match with the results obtained in [6] by conventional methods of NATD.

NATD as a solution generating transformation in Double Field Theory
The purpose of this section is to show that the NATD fields obtained by applying the transformation (2.6) and (3.22) where T in (2.4) is as in (3.17) are solutions of (generalised) supergravity equations. Since the O(d, d) transformation that produces the NATD fields is coordinate dependent, it is most useful to discuss this in the framework of Double field Theory (DFT), where O(d, d) arises as a manifest symmetry of the action and hence of the field equations. Therefore, we start with a brief review of DFT.

A Brief Review of Double Field Theory
DFT is a field theory defined on a doubled space, which implements the O(d, d) T-duality symmetry of string theory as a manifest symmetry. In addition to the standard space-time coordinates, the doubled space also includes dual coordinates, which are associated with the winding excitations of closed string theory on backgrounds with non-trivial cycles. The spacetime and dual coordinates transform as a vector under the T-duality group O(d, d): Herex In DFT, the dynamical fields are the generalised metric H, which is an element in SO − (d, d) that encodes the semi-Riemannian metric and the B-field; the generalised dilaton field; the spinor field S which is the element in Spin − (d, d) that projects onto H under the double covering homomorphism between Spin(d, d) and SO(d, d), that is ρ(S) = H. The spinor field χ encodes the RR fields in the democratic formulation of Type II supergravity. For more details see [35], [43].
The DFT action is as below: where and Here, is the Mukai pairing, which is a Spin(d, d) invariant bilinear form on the space of spinors [49]. This action has to be implemented by the following self-duality condition Moreover, one needs to impose the following O(d, d) covariant constraint, which is called the strong constraint: where A and B represent any fields or parameters of the theory. When the constraint is satisfied, the DFT action is gauge invariant under generalised diffeomorphisms and the gauge algebra closes under the C-bracket, which is an O(d, d) covariant extension of the Courant bracket. A special solution of the constraint is when none of the fields and gauge parameters have dependence on dual coordinates, that is, when∂ i = 0. In this case, the theory is said to be in the supergravity frame because for this solution of the constraint it can be shown that (4.3) reduces to the standard NS-NS action for the massless fields of string theory and (4.4) reduces to the RR sector of the democratic formulation of Type II supergravity theory.
The term R(H, d) in (4.3) is the generalized Ricci scalar and its explicit form is as follows: The DFT action presented in (4.4) is invariant under the following transformations: Here S ∈ Spin(d, d) and The dilaton field is invariant. The transformation rules for the generalized metric H = ρ(S) is determined by those of S and is as given below: The generalised Ricci scalar (4.7) is manifestly invariant under these transformations. A fact that is of crucial importance is that the transformation (4.10) is equivalent to [32] E where h = A B C D . (4.12) The transformation rule (4.9) for the fundamental fields in the theory make it possible to introduce the following anzats for the DFT fields: χ(X, Y ) = S(Y )χ(X) (4.14) Here S(Y ) belongs to the duality group Spin + (d, d).

Embedding NATD in Double Field Theory
We showed in the previous section that the NATD of a given Type II background with isometry G can be obtained through the action of the O(d, d) matrix (3.17) on it. As we have mentioned before, the field equations of DFT are equivalent to the field equations of Type II supergravity for the trivial solution of the constraint, that is when the fields are in the supergravity frame. As a result, the type II supergravity solution on which the NATD acts, can also be regarded as a a solution for DFT, for this trivial solution of the strong constraint. Now, assume that the isometry group G is unimodular 4 and that it acts freely on the background. This means that one can pick up coordinates in which the metric and the B-field can be written as in (3.1) and (3.2). We label these coordinates as {x 1 , · · · , x n−d , θ 1 , · · · , θ d }; then the dual coordinates will be labelled as {x 1 , · · · ,x n−d ,θ 1 , · · · ,θ d }. Obviously, the DFT fields H, S, d, χ that correspond to this background do not depend on the dual coordinates, that is, they are in the supergravity frame. Also, due to the existence of the isometry symmetry, the dependence of the generalised metric on the space-time coordinates is separated as follows (4.15) where L is the O(10, Our claim can be verified easily. Recall that the metric and the B-field can be written as in (3.1) and (3.2) with respect to our coordinate system. Let us consider the background matrix E = g + B. In this coordinate system This is equivalent to the following O(d, d) action (2.6): It can be verified easily that (4.18) is equivalent to (4.15), see [32]. 5 Similarly, the dependence of the field S, which is the Spin − (10, 10) element that projects onto H under the double covering homomorphism: ρ(S) = H and K ≡ C −1 S on the coordinates (x, θ) is also separated.
Here, S L is the P in(10, 10) matrix that projects onto L under the double covering homomorphism: ρ(S L ) = L. If we define K ≡ C −1 S, this implies We also assume that the p-form field strengths (not the gauge potentials) respect this isometry, that is, we assume that any p-form flux in the background can be written as where F µ 1 ···µni 1 ···i (p−n) (x) does not depend on any of the θ coordinates. In terms of the spinor field F that encodes these p-form field strengths, this means that F (x, θ) = S L (θ)F (x) (4.23) 5 Here, where F (x, θ) is the spinor field that encodes the field strengths written with respect to the coordinate basis. It can be shown that in this special case, whereL is as in (4.16), (4.23) is equivalent to the following 6 : As we have mentioned before, the above relation (4.24) is equivalent to Using the terminology from duality twisted (Scherk-Shwarz) reduction that we will discuss in subsection (4.3.1), we call the fields H(x), S(x), d(x) and F (x) the untwisted fields. Now, we apply the NATD transformation (2.6) and (3.22) where T in (2.4) is as in (3.17) on the untwisted fields. This will give us the dual fields H ′ , d ′ , S ′ and F ′ , which will depend on the coordinates {x 1 , · · · , x n−d , ν 1 , · · · , ν d }, which we collectively call {x, ν}.
where ρ(S T ) = T and B ′ (x, ν) is read off from the antisymmetric part of H ′ (x, ν) in (4.26).
Our strategy will be to show that these new fields H ′ (x, ν), d ′ (x, ν), S ′ (x, ν) and F ′ (x, ν) form a solution for the field equations of DFT in a frame in which the standard space-time coordinates are identified with {x 1 , · · · , x n−d , ν 1 , · · · , ν d } and the dual coordinates will be {x 1 , · · · ,x n−d ,ν 1 , · · · ,ν d }. As the field equations of DFT and Type II supergravity are equivalent in the supergravity frame, this will immediately ensure that the transformed fields are also a solution of Type II supergravity.
The key point in our argument will be that the two twist matrices L and T generate the same fluxes defined in the framework of Gauged Double Field Theory(GDFT). In the next section, we give a brief review of GDFT, and introduce the fluxes that arise in this context. Finally, we compute the fluxes associated with L and T and show that they are indeed the same.

Gauged Double Field Theory
GDFT is obtained from duality twisted (Scherk-Schwarz) reduction of DFT [40]- [43]. The O(d, d) invariance of the DFT action makes it possible to introduce the following Scherk-Schwarz type reduction anzats for the DFT fields: where ρ(S) = U −1 ∈ O(d, d) and F (x, Y ) = e B(x,Y ) / ∂χ(x, Y ). The matrices U and S are usually called twist matrices. When these anzatse are plugged into the action and gauge transformation rules of DFT, one ends up with GDFT, which is a consistent field theory, provided that the matrices U (Y ), S(Y ) satisfy a set of constraints, that we list shortly. The GDFT action is a deformation of the DFT action. In the NS-NS sector the Ricci scalar in (4.3) is deformed to with The anzats in (4.30) does not yield any deformation in the GDFT action of the RR sector, as it is F and not χ, which is twisted. As a result one ends up with the following action where v is defined as The integration is over x coordinates and their duals only and does not include the Y coordinates. In the action (4.34), the first two terms form the GDFT action for the NS-NS sector obtained by Scherk-Schwarz reduction of the DFT of NS-NS sector of Type strings. The fluxes that determine R f are the fluxes associated with the twist matrices U and S, whose general form we will give in the next subsection. The third term is the usual action for the RR sector of DFT of Type II strings and does not depend on the fluxes, as we have introduced the duality twisted anzats for F , which encode the RR fluxes rather the spinor field χ, which encode the modified gauge potentials. Once again, the relation between the two is F = e B / ∂χ. If it were the field χ which had been twisted, then the DFT action of the RR sector would have been deformed, too and the deformation would again be determined by the fluxes. It was shown in [43] that the Lagrangian in this case is of the same form as (4.4), where one replaces / ∂ with / ∇. Although we will not need this deformed action in this paper, we will need and present the explicit form of / ∇ in section (4.4), where we discuss the field equations arising from (4.34).

Fluxes, Dual Fluxes and O(d, d) invariance of GDFT
The fluxes that determine the deformation in the NS-NS sector are defined as below [42] f where σ is as in (4.31) and Note that Ω ABC are antisymmetric in the last two indices: Ω ABC = −Ω ACB . We also make the following definition The constraints that should be obeyed by the twist matrices are as follows: where g is any of the DFT fields (H, S, χ).

The DFT action of the NS-NS sector is manifestly
We have inserted ∂ and∂ in the arguments of R to emphasize that the derivatives ∂ M on the left hand side should be replaced by∂ N ≡ h M N ∂ M on the right hand side. On the other hand, the DFT action of the RR sector is Spin + (d, d) invariant [35]. Therefore, the field equations that arise from varying the DFT action of the NS-NS sector with respect to the generalised metric and the generalised dilaton are O(d, d) covariant, whereas the field equations obtained by varying the DFT action of the RR sector with respect to the spinor field χ or the spinor field S are covariant under the subgroup Spin + (d, d) of Spin(d, d). We will use this fact in section (4.4), see the equations (4.93, 4.94). P in(d, d) elements that do not lie in this subgroup act as dualities rather than invariances, as we will discuss more in detail in the next section. The O(d, d) invariance of the generalised scalar curvature R also extends to R f , provided that we treat the fluxes f ABC as spurious generalised tensors, which also transform under O(d, d). So, if we definef In this particular case, we call the resulting flux the dual flux, and we denote it byf ABC for this particular case. That is,f Note that, due to complete antisymmetry of f ABC in its indices, the only independent blocks of f ABC out of the 8 possible combinations are f a bc , f abc , f ab c and f abc , a = 1, · · · , d. It is customary to call these the geometric flux, the H-flux, the Q-flux and the R-flux, respectively [33]. The DFT fieldH we defined above has been called the dual generalised metric, in [35]. 7 Also note that we have ∂ i =∂ i ,∂ i =∂ i , that is the standard and dual derivatives have been swapped in ∂ and∂. For future reference, we also define the dual spinor fieldsF andχ as in [35]: It is easily checked thatF = eB / ∂χ. Here,B is the B-field associated with the dual generalised metricH.
At this point, it is natural to ask the relation of the twist matrixŪ associated with the fluxesf to the twist matrix U associated with the fluxes f . One can easily show that the relation isŪ = JU . Then,S = ±SC, where ρ(S) =Ū −1 and ρ(S) = U −1 . Indeed, it can easily be shown thatΩ For future reference, we also consider the fluxes associated with the twist matricesŨ = U J andS = ±CS, with ρ(S) =Ũ −1 . One can easily see that the fluxes associated withŨ are exactly the same as the fluxes associated with U , except for the fact that all standard/dual derivatives in the computation of f should be replaced with dual/standard derivatives in the computation off . More precisely, we havẽ

Fluxes associated with the NATD matrix
Let us now compute the fluxes associated with the matrices L in (4.16) and T in (3.17). Note that the condition are not included in the x coordinates of the fields H(x), d(x), χ(x) and S(x). In the computation, we take the coordinates on which the twist matrices depend as the standard coordinates and not the dual ones. Then For the matrix T in (3.17) we have where we have defined (so that the indices match) Plugging these in the formula we find that the only non-vanishing components are Ω a bc = C a bc , Ω abc = C e da C d bc ν e .
where the last equality follows from the Jacobi identity.
Now, let us compute the fluxes associated with the geometric twist matrix L (4.16). In this case we find that the only non-vanishing component is So, we have seen that the fluxes associated with T and L are exactly the same. This will be a key point in proving that NATD transformation produces solutions of GSE in the following subsection.

Comparing the Field Equations of DFT and GDFT
In the previous subsection, we studied GDFT, which is obtained from Scherk-Schwarz compactification of DFT. The Scherk-Schwarz anzats is known to give rise to a consistent dimensional reduction, meaning that any solution of the field equations of the resulting theory can be uplifted to a solution of the higher dimensional field equations [39]. In our case this implies that any solution of the field equations of the GDFT action can be uplifted to a solution for the DFT action in higher dimensions. Conversely, the field equations of DFT in higher dimensions will reduce to the field equations of GDFT in lower dimensions and hence, given a solution of DFT for which the dependence of the fields on the doubled coordinates is separated as in (4.29,4.30) and (4.31), the untwisted fields H(x), F (x), d(x) will form a solution of the GDFT, where the fluxes determining the GDFT is determined by the twist matrix U (Y ).
This straightforward argument should be discussed in more detail, mainly for two reasons. Firstly, the anzats in (4.29,4.30) and (4.31) is not exactly the Scherk-Scwarz anzats which gives a consistent dimensional reduction to GDFT, due to the difference in the anzats for RR fields. The correct anzats would have been which gives rise to a deformation of the RR sector, as well. Although the DFT action does only depend on F = e B / ∂χ, rather than the field χ itself, the gauge transformation rules have an explicit dependence on χ and hence a consistent reduction should involve an anzats for the field χ. 8 However, at the level of equations of motion, this raises no problems, as the equations only depend on / ∂χ = e −B F and has no dependence on χ (without the derivatives). As a second important point, the real duality group for DFT is Spin + (d, d) and hence only a twist matrix in this subgroup of Spin(d, d) can give a consistent reduction. This point is particularly important for us, as the NATD matrix in (3.17) is not in Spin + (d, d). However, as discussed in [35], although the P in(d, d) transformations which are not in this subgroup are not invariances of DFT, they act as duality transformations. This is also true at the level of field equations. In order to clarify these points, we will discuss below the relationship between the field equations of DFT and of GDFT in more detail.

Field Equations for the generalised dilaton field:
The field equations obtained by varying the DFT action with respect to the generalised dilaton field is [31] R = 0. (4.63) If we plug in (4.29) and (4.31) in (4.63), we obtain as was shown in [42]. It can easily be shown that this is the field equation obtained by varying (4.34) with respect to the generalised dilaton field. Therefore, a set of DFT fields whose dependence on the coordinates is separated as in (4.29) and (4.30) and (4.31) will satisfy the dilaton field equations of DFT if and only if the untwisted fields H(x), d(x) satisfy the dilaton equations for the GDFT, where the fluxes f ABC which determines the deformation is determined by the twsit matrix U .

Field Equations for the spinor field χ:
The field equation for the spinor field is [35] / ∂(K / ∂χ) = 0, (4.65) which is to be supplemented with the duality constraint In terms of the field F = e B / ∂χ the equation and the duality constraint becomes: Recalling that / ∂ = Γ M ∂ M we rewrite this equation as: Now we use the following facts [43] 9 where U = ρ(S −1 ). Using these one can show easily that the equation (4.72) is equivalent to where the Dirac operator / ∇ is defined as (see [43]) As a result, we conclude that the fields (4.29) and (4.30) form a solution for the field equation (4.69) if and only if the untwisted fields F (x), H(x) satisfy (4.76). 10 . 9 We proved the identity (4.74) in [43] for S ∈ Spin + (d, d). It can be easily shown that it also holds for elements of S ∈ Spin(d, d) of the form S = CS + and S = S + C, where S + ∈ Spin + (d, d). 10 Note that (4.76), which is equivalent to / ∇ / ∂χ(x) = 0 is not the field equation obtained from varying (4.34) with respect to the spinor field χ, which would have given / ∂ / ∂χ(x) = 0. It is not the field equation obtained from varying the GDFT action of the RR sector obtained in my paper through a duality twisted ansazt on χ (rather than F ) either, which would have yielded / ∇ / ∇χ(x) = 0. Note that both of these equations are satisfied automatically due to nilpotency of / ∂ and / ∇.

Field Equations for the generalised metric H M N :
The field equations obtained by varying the DFT action with respect to generalised metric H M N is [31], [35]: The first term in (4.78) comes from the variation of the GDFT action of the NS-NS sector, and the variation of the GDFT action of the RR sector gives the second term. Recall that F = e B / ∂χ. In passing to the second line in (4.79), we used the invariance property of Mukai pairing under Spin + (d, d) 11 (which e −B is an element of), and we also imposed the duality constraint / ∂χ = −K / ∂χ. Also, Γ M N is defined as Γ P Q ≡ 1 2 [Γ P , Γ Q ]. Let us plug in the set of fields in (4.29, 4.30, 4.31) into these equations. Consider first the following expression: Now, we plug in Ξ M N the fields H, K, F, B, whose dependence on the coordinates (x, Y ) is separated as in (4.29, 4.30, 4.31). If we use the invariance property of the Mukai pairing and the following identity we obtain S ∈ Spin ± (10, 10).
Therefore, we have found that Spin + (d, d). We assume that it is of the form S(Y ) = S 1 (Y )C, where C is the charge conjugation element satisfying ρ(C) = J. (This is the case for the twist matrices that determines our NATD fields. Recall that the NATD matrix is S T (ν) = S β (ν)C and S β ∈ Spin + (d, d).) In this case only the S 1 (Y ) factor can be dropped in passing from the first line to the second line in (4.87) and we end up with

Now consider the case when S is not in
where ρ(S 1 ) = U 1 andH = J T HJ is the dual generalised metric we defined in (4.49). Note that in writing the second line above we used Also recalling the definition of the dual spinor fieldF in (4.50) we see that (4.90) above can be written in terms of the dual fields and we have: Let us now try and write the R M N part of the field equation in terms of the dual generalised metricH, as well. For this we need to observe the following: wheref is the dual flux we defined in (4.47). These follow directly from (4.48). Recall that J is obtained by embedding the O(d, d) matrix J d in O(10, 10) as in (2.5), and hence acts on the partial derivatives along the directions x a , a = 1, · · · n − d as an identity transformation. Therefore, we have where g(x) denotes any of the GDFT fields H(x), F (x), d(x) or S(x). So, we have∂ = ∂ in (4.48). As a result, using (4.91), we can rewrite (4.84) as: To recap, we have obtained the following: For S ∈ Spin + (d, d) with ρ(S) = U −1 Here R f is calculated with respect to the fluxes corresponding to the twist matrix U . This then shows that the generalised metric field equations of DFT reduce to those of GDFT, as claimed.
On the other hand, if S = S 1 C with S ∈ Spin + (d, d) and ρ(S 1 ) = U −1 1 so that U = JU 1 we have and Rf is calculated with respect to the fluxes dual to f , which are associated with the matrix U . Therefore, comparing the field equations of DFT and the field equations of the resulting GDFT obtained by deforming it with a twist matrix U , which is not in SO + (d, d) is not straightforward, as it is when U ∈ SO + (d, d). In this case (which is also the case for the NATD fields), all we can say is the following: The fields H(x, Y ), F (x, Y ) and d(x, Y ) in (4.29-4.31) satisfy the generalised metric field equations of DFT if and only if the dual fieldsH(x),F (x) and d(x) satisfy the field equations (4.97).
We will discuss this further in the next subsection.

NATD Fields as Solutions of DFT in the Supergravity Frame
The main question we would like to answer is whether the NATD fields (4.26,4.27, 4.28) satisfy Type II field equations. Equivalently, we would like to answer whether these fields satisfy the field equations of DFT in a frame in which the coordinates (x, ν) are identified with standard space-time coordinates. This is because in the frame∂ i = 0 the DFT equations will reduce to Type IIA or Type IIB equations depending on the fixed chirality of χ.
In the previous section, we saw that the fields H(x, Y ), F (x, Y ), S(x, Y ) and d(x, Y ) whose dependence on x and Y coordinates is separated and the Y dependence is determined by a twist matrix in SO + (d, d) satisfy the field equations of DFT if and only if the untwisted fields H(x), F (x), S(x) and d(x) satisfy the field equations of the GDFT determined by the fluxes associated with the twist matrix. This immediately implies the following: Suppose that we know the fields H(x, Y ), F (x, Y ), S(x, Y ) and d(x, Y ) satisfy the field equations of DFT. This implies that the untwisted fields satisfy the field equations of GDFT determined by the fluxes associated with U and S. Now, consider another set of fields H 1 (x, Z), F 1 (x, Z), d 1 (x, Z) obtained by twisting the same fields H(x), F (x), d(x) by the twist matrices U 1 (Z) and S 1 (Z), where U 1 is also in SO + (d, d). Suppose also that the fluxes generated by U 1 (Z) and S 1 (Z) are the same as the fluxes generated by U (Z) and S(Z). Since we already know that the untwisted fields satisfy the field equations of GDFT determined by these fluxes, we immediately conclude that the twisted fields H 1 (x, Z), F 1 (x, Z), d 1 (x, Z) satisfy the field equations of DFT, too. This is when U, U 1 ∈ SO + (d, d).
On the other hand, we saw that comparing the generalised metric field equations of DFT and the GDFT is subtle, when the twist matrix U determining the deformation is not in SO + (d, d). The subtlety arises due to the fact that the DFT of the RR sector of Type II strings is invariant only under the subgroup Spin + (d, d) and P in(d, d) transformations that are not in this subgroup must be viewed as dualities and not invariances. In analyzing this case, we found it useful to define the following dual variables, as in [35]. Let us write them down here again (4.98) Recall thatF = eB / ∂χ, whereχ = Cχ.
It is possible to formulate the DFT action in terms of these dual variables. In fact, it was shown in [35] that the DFT action takes the same form in terms of these dual fields as the action (4.4), provided that we also transform the partial derivatives as ∂ i ↔∂ i , i = 1, · · · , d. We will call this action the dual DFT action. 12 If the chirality of the spinor field χ is fixed such that the DFT action reduces to the action of Type IIA/IIB theory in the supergravity framẽ ∂ i = 0, the dual DFT action reduces to the action of Type IIB/IIA theory in the frame ∂ i = 0, if d is odd, cite Zwiebach, Hull, etc. If d is even, the chirality of the dual spinor field remains the same, hence the dual action reduces to the same Type II action in the frame ∂ i = 0. 13 Consider a set of fields, which form a solution for the DFT field equations in a certain frame. Then, the dual fields will satisfy the equations arising from the DFT action written in terms of the dual fields. We emphasize again that these equations have exactly the same form as the equations in terms of the original fields, except that the derivatives along the directions on which J d acts have been replaced by the dual derivatives and vice versa. Now suppose that the frame in which the fields satisfy the DFT equations is the supergravity frame (that is, the fields have no dependence on dual coordinatesx 1 , · · · ,x d ). As a result, these fields form a solution of Type IIA(/IIB) supergravity. Since the dual fields will not belong to the frame ∂ i = 0 in general, they do not necessarily form a solution of Type IIB(/IIA) supergravity. Nevertheless, they are a solution of the field equations of the dual DFT action, and that is all the information we need. 14 If the dependence of the fields forming the DFT solution on the coordinates (x, Y ) is separated as in (4.29,4.30,4.31), then the dependence of the dual fields on these coordinates is also separated in the following way: 101) 12 In fact, the DFT action of the RR sector picks up an overall minus sign but so does the duality condition.
Hence, when we plug in the duality condition into the field equations, there is no overall minus sign and the form of the field equations are exactly the same both in terms of the original and the dual fields and coordinates. 13 If the time direction is also dualized, the resulting theory is Type IIA ⋆ or Type IIB ⋆ depending on the chirality, see [35]. 14 If the solution has an isometry symmetry U (1) d so that one can pick up coordinates with respect to which the twisted fields will have no dependence on the coordinates x i , i = 1, · · · , d as well, the dual fields will belong to the frame ∂i = 0. Being a solution of the dual DFT equations, they will hence form a solution of Type IIB(/IIA) supergravity. This is what happens in Abelian T-duality. whereŜ = CSC −1 , ρ(Ŝ) =Û −1 = JU −1 J, andF (x, Y ) = eB (x,Y ) C / ∂χ(x, Y ) andd = d. Because the equations that these dual fields satisfy are exactly of the same form as the generalised metric field equations we discussed above [35], we have (as S ∈ Spin + (d, d), we also havê S = CSC −1 ∈ Spin + (d, d)): where the fluxesf in Rf is calculated with respect to the twist matrixÛ (Y ) and in calculating the fluxes one should replace ∂ i ↔∂ i . Now remember our discussion in section (4.3.2). From (4.52) we immediately see that these are just the fluxes produced by the matrixŨ =Û J. Sincê U = JU J we see that the fluxesf are the same fluxes as those produced by the twist matrix U =Û J = JU , since J 2 = Id. But, according to (4.51) this is just the dual flux to the flux f produced by the twist matrix U . As a result we see,f =f . Now, recall that the DFT fields constructed out of a Type II supergravity solution with isometry group G satisfies the following set of equations where f is the geometric flux with f c ab = C c ab , the latter being the structure constants of the Lie algebra of G with respect to a fixed basis. According to our discussion above, the dual fields satisfy the following equations where the fluxf is the dual of the geometric flux, that is, the only non-vanishing part is the Q-flux part: f ab c = δ ad δ be δ cf C f de . Then, we see from (4.97) that we have for any set of DFT fields H(x, Y ), F (x, Y ), d(x, Y ), if they are obtained by twisting the fields H(x), F (x), d(x) by a twist matrix U , which also produces the geometric flux with f c ab = C c ab . But, we know that the NATD matrix (3.17) that generates the NATD fields (4.26-4.28) produces exactly the same fluxes. This then proves that the NATD fields (4.26-4.28) indeed from a solution to the generalised metric field equations of DFT in a frame in which the coordinates (x, ν) are identified by the the space-time coordinates. In a similar way, one also sees that the other field equations coming from the variation of the DFT action with respect to the generalised dilaton field d and the spinor field χ are also satisfied by the NATD fields (4.26-4.28), as we know that they are satisfied by the fields H(x, θ), F (x, θ) in (4.15), (4.24), since both sets of fields are obtained via twist matrices (namely, L and T ), which produce exactly the same fluxes. Since we identify the coordinates (x, ν) with the space-time coordinates, we are in the supergravity frame where the DFT field equations reduce to the field equations of Type II supergravity. This completes the proof that the NATD fields form a solution of the field equations of Type II supergravity.

Non-Unimodular Case: Generalised Supergravity Equations
So far, we have assumed that the isometry group G is unimodular, and hence the structure constants C c ab is traceless. Then, the geometric flux associated with the twist matrix L, and the dual Q-flux are also traceless. When this assumption is relaxed, it is known that the resulting NATD fields form a solution of the GSE, which have recently been introduced in [18], [19]. Let us see see how this arises within the framework of DFT we discuss here.
For simplicity, we assume that the structure constants of the Lie algebra of G have only trace components. Then, the only non-vanishing components of the flux associated with the twist matrix L in (4.15) will be f a , a = 1, · · · , d. This contributes to η a , whose definition is given in (4.36). However, it is well-known that the GDFT action with non-vanishing η A is not consistent [40,42]). Therefore, the f a part in (4.36) should be compensated by a non-trivial dilaton anzats. A similar situation was also considered [50]. Rewriting (4.36) in components, we see that we need to have In other words, σ is linear in the dual coordinates and does not depend on the standard coordinates. Then, the generalised dilaton field will have a linear dependence on the dual coordinates as follows: where m i are constants.
Due to the dependence of the dilaton field on the dual coordinates, we are not in the supergravity frame anymore. Also note that, due to the form of the anzats (4.30), the spinor field F also has a dependence on the dual coordinates. The other DFT fields H and S depend only on the space-time coordinates. This frame in which the NATD fields satisfy the DFT equations was studied in the papers [44] and [45]. In those papers, they have shown that the equations of DFT in such a frame reduce to GSE. As a result, the NATD fields form a solution of GSE, when G is not unimodular.

Conclusions and Outlook
In this paper, we studied NATD as a coordinate dependent O(d, d) transformation. The dependence of the NATD matrix is on the coordinates of the dual theory, and is determined by the structure constants of the Lie algebra of the isometry group G. Besides making calculations significantly easier, our approach gives a natural embedding of NATD in Double Field Theory (DFT), which provides an O(d,d) covariant formulation for effective string actions [25]- [35] by introducing dual, winding type coordinates. As a result of this embedding, we managed to give a simple proof of the fact that the NATD fields solve supergravity equations, when the isometry algebra is unimodular. When the isometry algebra is non-unimodular, we showed that the generalised dilaton field of DFT is forced to have a linear dependence on the dual coordinates, which then meant that the resulting background solved GSE.
We believe that identifying the O(d, d) matrix that generates the NATD background is rather important, as it should make it easier to study some properties of the NATD backgrounds and their CFT duals (such as supersymmetry and integrability), as the relation to the original background is more explicit. On the other hand, our approach also makes it possible to explore the relation between NATD and Yang-Baxter (YB) deformations in detail. Homogoneous YB deformation of an integrable sigma model [51] is determined by the so called r-matrix, which forms a solution of the classical Yang-Baxter equation. In the paper [52], it was conjectured that homogoneos YB models can be obtained by applying Non-Abelian T-duality (NATD) to the original background, with respect to an isometry group determined by the r-matrix. This conjecture was proved in [53] for the case of Principal Chiral Models (PCM) and they extended their work to homogenous YB deformations of more general sigma model than PCM's in [17]. Then, the results of our paper implies that it should be possible to describe YB deformations also as O(d, d) transformations. Indeed, this was shown to be the case in the works [54]- [57]. The methods we have developed in this paper should give deeper insight on YB deformations and the relation between NATD and YB deformations. We hope to come back to these issues in the near future [58].