Type II DFT solutions from Poisson-Lie T-duality/plurality

String theory has the T-duality symmetry when the target space has Abelian isometries. A generalization of the T-duality, where the isometry group is non-Abelian, is known as the non-Abelian T-duality, and it works well as a solution generating technique in supergravity. In this paper, we describe the non-Abelian T-duality as a kind of O(D,D) transformation when the isometry group acts without isotropy. We then provide a duality transformation rule for the Ramond-Ramond fields by using the technique of double field theory (DFT). We also study a more general class of solution generating technique, the Poisson-Lie (PL) T-duality or T-plurality. We describe the PL T-plurality as an SO(D,D) transformation and clearly show the covariance of the DFT equations of motion by using the gauged DFT. We further discuss the PL T-plurality with spectator fields, and study an application to the AdS$_5\times$S$^5$ solution. The dilaton puzzle known in the context of the PL T-plurality is resolved with the help of DFT.

equations of motion [23,25,28]. However, as pointed out in [142,143], the dual geometry in fact satisfies the generalized supergravity equations of motion (GSE) [144,145]. When the target space satisfies the GSE, string theory has the scale invariance [144,146] and the κ-symmetry [145]. The conformal symmetry may be broken, but recently, a local counterterm that cancels out the Weyl anomaly was constructed in [147] (see also [148]), and string theory may be consistently defined even in the generalized background. Even if it is not the case, NATD for a non-unimodular algebra still works as a solution generating technique in supergravity, because an arbitrary GSE solution can be mapped to a solution of the usual supergravity [144,[148][149][150] by performing a (formal) T -duality. Then, combining the NATD with f ab a = 0 and the formal T -duality, we can generate a new supergravity solution.
We also study the PL T -plurality with the R-R fields. In fact, the PL T -plurality can be regarded as a constant SO(D, D) transformation acting on "untwisted fields" {Ĥ AB ,d,F} .
By requiring the untwisted fields to satisfy the dualizability condition or the E-model condition of [147], we show that the DFT equations of motion in the original and the transformed background are covariantly related by the SO(D, D) transformation. This shows that the if the original background satisfies the DFT equations of motion, the transformed background also is a solution of DFT. We also discuss the PL T -plurality with spectator fields. Again, requiring a certain T -dualizability condition, we show the dilaton equation of motion is satisfied in the dual background, although the full equations of motion are not checked. By using the duality rules, we study an example of the PL T -plurality with the R-R fields.
Under a PL T -plurality transformation, a dual-coordinate dependence (i.e. dependence on the coordinates of the dual groupG) can appear in the dilaton. When such coordinate dependence appears, the background does not have the usual supergravity interpretation, and we are forced to disallow such transformation. However, in DFT, we can treat the dual coordinates and the usual coordinates on an equal footing and we do not need to worry about the dilaton puzzle.
As discussed in [148,150], a DFT solution with a dual-coordinate dependent dilaton can be regarded as a solution of GSE, and by performing a further formal T -duality, we can obtain a linear dilaton solution of the usual supergravity. In this way, the issue of the dilaton puzzle is totally resolved and we can consider an arbitrary T -plurality transformation.
This paper is organized as follows. In section 2, we briefly review DFT and GSE. In section 3, we begin with a review of the traditional NATD, and translate the results into the language of DFT. We then provide a general transformation rule for the R-R fields. Examples of NATD without and with the R-R fields are studied in section 4 and 5. In section 6, we study the PL T -plurality in terms of DFT and determine the transformation rules from the DFT equations of motion. As an example of the PL T -plurality, in section 7, we study the PL T -plurality transformation of AdS 5 × S 5 solution. Section 8 is devoted to conclusions and discussions.

A review of DFT and GSE
Generalized-metric formulation of DFT There are several equivalent formulations of DFT, but the generalized-metric formulation [119,120,122,126] may be most accessible one, and in this paper, we utilize this formulation as much as possible. In this formulation, the fundamental fields are a symmetric tensor, called the generalized metric H M N (x) , and a scalar density e −2 d(x) called the DFT dilaton. The Lagrangian is given by In the generalized-metric formulation, for consistency, we assume that arbitrary fields or gauge parameters A(x) and B(x) satisfy the so-called section condition, According to this requirement, all of the fields cannot depend on more than D coordinates.
Under the section condition, the DFT action is invariant under the generalized Lie derivativê 6) and this generates the gauge symmetry of DFT, known as the generalized diffeomorphisms.
Under the section condition, we can also check that the generalized Lie derivative is closed by (2.7) In particular when the inner product of V M a is constant, η M N V M a V N b = 2 c ab (c ab : constant) , we can easily show that the C-bracket coincides with the generalized Lie derivative (2.10) If we use the curvature tensor, the invariance of the DFT action under the generalized diffeomorphism is manifest. In other words, the DFT action can be understood as a natural generalization of the Einsten-Hilbert action.
The equations of motion are also summarized in a covariant form as 1 where we the generalized Ricci tensor is defined by For concrete computation, the following expression may be more useful: (2.14) When we make the connection to the conventional supergravity, we remove the dependence on the dual coordinates∂ m = 0 and parameterize the generalized metric and the dilaton as by using the standard NS-NS fields (g mn , B mn , Φ) . Then, S and S M N reduce to 16) and the standard supergravity Lagrangian We can also introduce the R-R fields in a manifestly O(D, D) covariant manner. However, the treatment of the R-R fields is slightly involved, and we will not write down the covariant expression explicitly here (for the detail, see for example [148,152], which is consistent with our conventions). In the following, aimed at readers who are not familiar with DFT, we will try to describe the R-R fields as the usual p-form fields as much as possible.

Gauged DFT
When we manifest the covariance under the PL T -plurality, it is convenient to rewrite the DFT equations of motion (2.11) by using a technique of the gauged DFT [130][131][132][133].
Suppose that the generalized metric H M N has the form, where theĤ AB is a constant matrix, which we call the untwisted metric. In this case, it is useful to define F ABC and F A , which are called the gaugings or the generalized fluxes, as 2 (2.20) which behave as scalars under generalized diffeomorphisms.
By using the generalized fluxes, we can show that the DFT equations of motion (2.11), under the section condition, are equivalent to where (2.22) Here, we have defined

24)
P ABCDEF ≡P ADP BEP CF + P ADP BEP CF +P AD P BEP CF +P ADP BE P CF 25) and the indices A, B are raised or lowered with η AB and η AB . Under the section condition, we can check that R = S . The equivalence between S M N = 0 and G AB = 0 is slightly more non-trivial but it is concisely explained in [133].
In the flux formulation of DFT [133], we take the untwisted metricĤ AB as a diagonal Minkowski metric, and then E A M ≡ U A M is regarded as the generalized vielbein. The fundamental fields are E M A and d , and the equations of motion (2.21) can be derived from On the other hand, in this paper, we rather interpret (2.19) as a reduction ansatz and the equations of motion (2.21) are just rewritings of (2.14), similar to the gauged DFT [130][131][132].
For our purpose, it is enough to consider the cases where the generalized fluxes are constant. 2 The convention for FABC is opposite to the standard one.
In that case, the equations of motion are simple algebraic equations where we have again used the section condition.
In general, the untwisted metric and the DFT dilaton may depend on the coordinates y µ on the uncompactified external spacetime. In this case, we denote the extended coordinates as (x M ) = (y µ , x i ,ỹ µ ,x i ) and consider By following [132], we assume thatĤ AB (y µ ) andd satisfies and then the generalized Ricci scalar becomes [132] S =Ŝ + 1 12

GSE from DFT
As we already explained, if we choose a section∂ m = 0 , the DFT equations of motion reproduce the usual supergravity equations of motion. On the other hand, we can derive the GSE by choosing another solution of the section condition [148,150], where the DFT dilaton has a linear dependence on the dual coordinates. In order to satisfy the section condition, we require the I m to satisfŷ These are equivalent to and indeed ensure the section condition, If we take this section and parameterize H M N as usual in terms of (g mn , B mn ) and d 0 as where U m ≡ I n B nm . They are precisely the GSE studied in [144][145][146]. When I m = 0 (where the Killing equations are trivial), they reduce to the usual supergravity equations of motion.
Another way to derive the GSE is to make a modification everywhere in the DFT equations of motion [150]. As long as the X M satisfieŝ we can choose a gauge such that X M takes the form (2.34) [148]. In terms of the generalized flux, obviously, this modification corresponds to Even in the presence of the R-R fields, this replacement is enough to derive the type II GSE, although we additionally need to require the isometry condition for the R-R fields, In generalized backgrounds, where the supergravity fields satisfy the GSE, the string theory may not have the conformal symmetry. Accordingly, when we obtain a generalized background as a result of NATD, it is usually regarded as a problematic example, and such background is not considered seriously. However, as discussed in [144,[148][149][150], by performing a formal T -duality, we can always transform a generalized background to a linear-dilaton solution of the usual supergravity. Here, we review what is the formal T -duality.
The DFT equations of motion are covariant under a constant O(D, D) transformation, In particular, if we consider an O(D, D) matrix, The reason why we call this O(D, D) transformation a "formal" T -duality is as follows.
The usual Abelain T -duality is a O(D, D) transformation,

Non-Abelian T -duality
In this section, we study the traditional NATD in general curved backgrounds. We begin with a review of NATD for the NS-NS sector. We then describe the duality as a kind of O(D, D) rotation and provide the general transformation rule for the R-R fields by employing the results of DFT. To provide a closed-form expression for the duality rule, we restrict our discussion to the case where we can take a simple gauge choice, x i (σ) = const.

NS-NS sector
In the case of the Abelian T -duality, the dual action can be obtained by following the procedure of [8,11]. When a target space has a set of Killing vectors v m a that commute with each other [v a , v b ] = 0 , the sigma model has a global symmetry generated by x m (σ) → x m (σ) + ǫ v m a (σ) . This global symmetry can be made a local symmetry by introducing gauge fields A a (σ) and We also introduce the Lagrange multipliersx a (σ), which constrain the field strengths to vanish. Then, by integrating out the gauge fields A a , we obtain the dual action, where the Lagrange multipliersx a becomes the embedding function in the dual geometry. In [19], this procedure was generalized to the case of non-commuting Killing vectors. It was further developed later, and in the following, we review NATD in a general setup discussed in [25,35].
We consider a target space with n generalized Killing vectors V a (a = 1, . . . , n) satisfyinĝ Here, c ab is a constant symmetric matrix. If we choose a section∂ m = 0 and parameterize the generalized Killing vectors as these conditions reduce to where the dot denotes a contraction of the index m . They are precisely the requirements to perform NATD [25,35] (see [153,154] for the origin of the conditions). 3 Under the setup, we consider the gauged action by following the standard procedure [8,11].
Ignoring the dilaton term, the gauged action takes the form [25,35,153,154] where we have introduced gauge fields A a (σ) ≡ A a a (σ) dσ a (a = 0, 1) and have defined Under the conditions (3.1), this action is invariant under the local symmetry, If we first use the equations of motion for the Lagrange multipliersx a , the field strengths F a are constrained to vanish and the gauge fields will become a pure gauge. Then, at least locally, we can choose a gauge A a = 0 and the original theory will be recovered, On the other hand, by using the equations of motion for A a first, we obtain the dual model.
For this purpose, it is convenient to rewrite the action as Then, the equations of motion for A a become 4 and this can be solved for A a as where we have defined After eliminating the gauge fields, the action becomes where E mn ≡ g mn + B mn . In the above computation, we have assumed that the matrix (E ab + f ab cx c ) is invertible, 5 but other than that the computation is general. Now, a major difference from the Abelian case appears. In the Abelian case, by choosing the adapted coordinates v m a = δ m a , we can always realize a gauge x a (σ) = 0 . However, in the non-Abelian case, such gauge choice is not always possible since we cannot realize v m a = δ m a . In order to provide a closed-form expression for the duality transformation rule, in this paper, we assume that the gauge symmetries can be fixed as x i (σ) = c i (c i :constant) under a suitable decomposition of spacetime coordinates (x m ) = (y µ , x i ) . This gauge choice removes n coordinates and instead n dual coordinatesx a are introduced. Then, the situation is the same as the Abelian case.
Under the gauge choice x i (σ) = c i , the action (3.13) reproduces the dual action for the dual coordinates x ′m = (y µ ,x a ) , 14) Then, the NATD can be understood as a transformation of the target space geometry, Regarding the transformation rule for the dilaton, we employ the result of [19],

NATD as O(D, D) transformation
In order to show a general transformation rule for the R-R fields, it is convenient to describe NATD as O(D, D) rotations. Starting with the original background, we construct the dual background (3.15) through the following three steps.
1. We first perform a GL(D) transformation, As we assumed, we can fix the gauge symmetry δ ǫ x i = ǫ a v i a such that x i (σ) = c i is realized. For this to be possible, det(v i a ) = 0 should be satisfied and the matrix Λ v is invertible. We then obtain 2. We next perform a B-transformation and obtain 3. Finally, we perform a T -duality transformation, and obtain the determinant of the metric transforms as (see for example [155]) Therefore, under the NATD (3.26), we obtain Combining this with (3.17), we obtain the transformation rule for the DFT dilaton

R-R sector
This shows that the R-R fields also should transform covariantly, in order to satisfy the equations of motion of type II DFT, where E M N is an O(D, D)-covariant energy-momentum tensor made of the R-R fields.
In DFT, there are basically two approaches to describe the R-R fields. One treats the R-R fields as an O(D, D) spinor [124], which is based on the earlier work [156], and the other treats them as an O(D) × O(D) bi-spinor [127], which is based on the approach of [157,158].

R-R fields as a polyform
We first explain the former because it is simpler. Since the treatment of O(D, D) spinor can be rephrased in terms of the differential form, here we treat the R-R field strength as the usual polyform (we follow the convention of [148]), a polyform F (that corresponds to an O(D, D) spinor) transforms as a GL(D) tensor, 2. Under the B-transformation, a polyform F transforms as 3. Under the (factorized) T -duality along the x m -direction, it transforms as wherex m is the coordinate dual to x m , and ∨dx m denotes the interior product acting from the right.
4. An arbitrary O(D, D) transformation can be decomposed into the above three types of transformations, but for later convenience, we also show that under the β-transformation, the transformation rule is given by By using the rules, the general formula for R-R fields under the NATD (3.26) becomes where the order of T y 1 · · · T y n is not important since the overall sign flip is a trivial symmetry.
Note that the field strength F = dA is known as the field strength in the A-basis [159] (which is sometimes called the Page form). Another definition, G ≡ dC + H 3 ∧ C , is known as the C-basis (see Appendix (A)). In the dual background, G can be obtained as We note that the approach of [80] based on the Fourier-Mukai transformation (see also [96] for an application) will be closely related to the procedure explained here.

R-R fields as a bi-spinor
Next, let us also explain the treatment of the R-R fields as a bi-spinor G α β . Starting with a polyform G , by using a vielbein e m a associated with g mn , we define the flat components G a 1 ···ap = e m 1 a 1 · · · e mp ap G m 1 ···mp and then define the bi-spinor G as where Γ a 1 ···ap ≡ Γ [a 1 · · · Γ ap] and (Γ a ) α β is the usual gamma matrix satisfying {Γ a , Γ b } = 2 η ab . According to [127,157,158] (see also [152]), under a general O(D, D) rotation the bi-spinor transforms as where Ω is a spinorial representation of the Lorentz transformation Λ a b , andΓ a ≡ Γ 11 Γ a . In particular, under a T -duality along a (spatial) x z -direction, we have When the vielbein has a diagonal form e a m ∝ δ z m , Ω is just the gamma matrix Ω = Γ z . The Ω corresponding to the β-transformation was obtained in [152] as where AE is similar to an exponential function defined in [157] AE 1 2 β ab Γ ab ≡ 5 p=0 1 2 p p! β a 1 a 2 · · · β a 2p−1 a 2p Γ a 1 ···a 2p , (3.50) where the position of the indices a, b are changed with the flat metric η ab , and we also defined (3.51) Now, let us consider the NATD (3.26). Since it is not easy to find a general expression for Ω , let us truncate the B-field and restrict assume a simple background, We also suppose the generalized Killing vectors are given by and using this, we define the flat components and the bi-spinor as Under the first GL(D) transformation, G is invariant while the internal part of the vielbein becomes an identity matrix e a i = δ a i . We next perform the B-transformation and T -dualities, but it is useful to perform the T -dualities first because the vielbein is now just an identity matrix. Namely, we rewrite the B-transformation and T -dualities as T -dualities and the β-transformation with parameter χ ab ≡ f ab cx c , Under this transformation, the bispinor is transformed as This appears to be consistent with the formula given in eq. (3.8) of [73] up to convention.
If we need to consider the spacetime fermions such as the gravitino and the dilatino, they are also transformed by this Ω , and this approach will be important. However, in order to determine the transformation rule for the R-R fields, the first approach will be more useful.

Examples without R-R fields
In this section, we study examples of NATD without the R-R fields. In the absence of the R-R fields, our setup is basically the same as the standard one. In order to find new solutions, we consider NATD for non-unimodular algebras f ba b = 0 .
As found in [23], in non-unimodular cases, the dual geometry does not solve the supergravity equations of motion. However, as recently found in [142], the dual geometry is a solution of GSE. Additional examples were discussed in [143], and there, by using the result of [28], it was shown that the Killing vector I in GSE is given by a simple formula, As we reviewed in section 2, an arbitrary solution of GSE can be regarded as a solution of DFT with linear dual-coordinate dependence. Through a formal T -duality, this can be mapped to a solution of the conventional supergravity. In this section, we generate new solutions of supergravity by combining the NATD for a non-unimodular algebra and the formal T -duality.
In fact, by allowing for non-unimodular algebras, we can perform a rich variety of NATD.
In order to demonstrate that, we consider several non-Abelian T dualities of a single solution, the AdS 3 × S 3 × T 4 background with the H-flux.

AdS 3 × S 3 × T 4 : Example 1
In the first example, we introduce the coordinates as We then consider the generalized isometries generated by The structure constant has the non-vanishing trace f b2 b = f 12 1 = 1 and the dual background will be a solution of GSE.
The B-field is not isometric along the v 1 direction £ v 1 B 2 = 0 , and the dual componentṽ 1 is necessary to satisfy the generalized Killing equations £ v 1 B 2 + dṽ 1 = 0 . Moreover, in order to realize [V 1 , V 2 ] C = V 1 , the dual component of V 2 is also necessary. In this case, we find but the requirement f ab d c dc = 0 in (3.1) is not violated and we can perform the NATD. The gauge symmetry associated with the generalized Killing vector V 2 can be fixed by realizing x + (σ) = 1 . Similarly, the gauge symmetry associated with V 1 can be also fixed as z(σ) = 1 .
The AdS parts of the matrices in the formula (3.26) (before the gauge fixing) become and under the gauge x + = 1 and z = 1 , the dual background becomes (4.8) As expected, this background does not solve the conventional supergravity equations of motion, but instead satisfies the GSE with the Killing vector Interestingly, this geometry is locally the same as the original AdS 3 ×S 3 spacetime. Indeed, by changing coordinates as we obtain the expression (4.11) In fact, we can find a 2-parameter family of solutions 12) and the NATD connects the original background (c 0 , c 1 ) = (0, 0) and the dual (c 0 , c 1 ) = (1, 1).
The metric in (4.11) is the same as the original one (4.2) and the B-field is also just shifted by a closed form B 2 → B 2 + 2 dx + ∧ d ln z . The only difference from the original background is in the dilaton. We note that, unlike the case of "trivial solutions" [152], we cannot remove the Killing vector I m in the dual geometry (4.11). 6 It will be natural to consider performing a B-field gauge transformation in order to undo the shift in the B-field. However, in the conventional GSE, where the only modification is given by the Killing vector I m , the gauge symmetry for the B-field is already fixed and we cannot perform a B-field gauge transformation. Indeed, if we truncate the closed-form in the B-field by hand, we find another solution where c 0 is a free parameter and c 1 can take two values, c 1 = 0 or c 1 = 1 . This is an example of the trivial solution and c 0 can be chosen as c 0 = 0 . Then, we get two AdS 3 × S 3 × T 4 solutions of the supergravity, with a different dilaton c 1 = 0 or c 1 = 1 .
For an arbitrary GSE solution, by taking a coordinate system with if we perform a formal T -duality that exchangesx z with the physical coordinate x z , we can get a solution of the conventional supergravity where the DFT dilaton is d = d 0 + I z x z . In the present example (4.12), we perform a formal T -duality along the x + -direction, and then the DFT dilaton becomes a function of the physical coordinates (4.14) Then, the dual-coordinate dependence disappears from the background fields. However, in this case, the AdS part of the dualized generalized metric becomes and we cannot extract the supergravity fields (g mn , B mn , Φ) from H M N due to det(g mn ) = 0 .
This type of (genuinely) DFT solution is called the non-Riemannian background [161], which is studied in detail in [162][163][164][165]. Using a parameterization given in [163], we find (4. 16) In the parameterization of [163], in general, there are n pairs of vectors (X i , Y i ) andñ pairs of vectors (Xī,Ȳī) , and such a non-Riemannian background is called a (n,ñ) solution. In this notation, this background is a (1,1) solution.
In this way, in the first example of NATD, the formal T -duality does not produce the usual supergravity solution, and we instead obtained a (1,1) non-Riemannian background.

AdS 3 × S 3 × T 4 : Example 2
In the second example, we take the coordinates, and consider the translation and the dilatation generators as the generalized Killing vectors, Here, we fix the gauge as x(σ) = 0 and z(σ) = 1 .
The AdS 3 parts of the transformation matrices are 19) and the NATD gives (4.20) This satisfies the GSE by introducing the Killing vector as Again, in order to remove the Killing vector I , let us perform a formal T -duality along thez-direction. This yields a simple linear-dilaton solution of the supergravity, where the AdS part of the B-field has disappeared.

AdS 3 × S 3 × T 4 : Example 3
We next use the Rindler-type coordinates, and consider the generalized Killing vectors Here, we take a gauge t(σ) = 0 and x(σ) = 1 .
The AdS parts of the transformation matrices are (4.24) and the dual background, which satisfies the GSE, becomes (4.25) In order to obtain a solution of the supergravity, if we again perform a formal T -duality along the time directiont . This time again we find a non-Riemannian background, where the S 3 × T 4 part of the generalized metric is not displayed. This is also a (1,1) solution, To summarize shortly, NATD works well a solution generating technique of DFT even if the isometry algebra is non-unimodular. If we additionally perform a formal T -duality, we usually obtain the usual supergravity solution, but sometimes, the parameterization of the generalized metric becomes singular, and we obtain a non-Riemannian background. The non-Riemannian backgrounds do not have the usual supergravity interpretation, but they have interesting applications [162][163][164][165] and they are interesting backgrounds by their selves.
Therefore, it will be important to study NATD for non-unimodular algebras more seriously.

Examples with R-R fields
In this section, we consider NATD with the R-R fields. After reproducing a known example, again we consider examples for non-unimodular algebras.
For convenience, let us display the summary of the duality rules. Under the setup, , the dual background is given by and the coordinates are transformed as (

AdS
As the first example of NATD with the R-R fields, let us review the example of [72] and demonstrate that our formula gives the same result. The original background is where AdS 3 and S 3 part have the curvature R = ∓6 ℓ 2 .
We perform the non-Abelian T -dualities associated with three generalized Killing vectors on the S 3 , which satisfy As clear from the explicit form of the Killing vectors, we can choose a gauge The (θ, φ, ψ) parts of the transformation matrices are 8) and the NS-NS fields in the dual background are Now, let us consider the R-R fields. Under the gauge (5.7), the Page form becomes The first GL(D) transformation is trivial Λ v = 1 under the gauge (5.7). We next perform the Finally, by performing T -dualities along (θ, φ, ψ)-directions, we obtain From this Page form, we get the R-R field strengths in the C-basis as (5.13) They are precisely the same as the results of [72] (where ℓ = 1/2).
Since the R-R potential also behaves an O(D, D) spinor in DFT, let us also explain how to determine the R-R potential in the dual background. Due to the gauge fixing (5.7), the Page form takes the form (5.10). Then the R-R potential in the A-basis is where θ should not be set to θ = π/2 in order to realize F = dA . Similar to the field strength, where we denotedũ 1 ≡ θ as it is dual to u 1 =θ . As A depends on the dual coordinate explicitly, the relation between F and A is generalized as 7 16) and the A ′ in (5.15) correctly reproduces the F ′ obtained in (5.12). This result is consistent with [125] where the massive IIA supergravity was reproduced from DFT by introducing a linear dual-coordinate dependence into the R-R 1-form potential. The potential in the C-basis also can be obtained by computing

AdS 5 × S 5
As the second example, let us consider a NATD of the AdS 5 × S 5 background associated with a non-unimodular algebra. The original background is where ds 2 S 5 ≡ dr 2 + sin 2 r dξ 2 + sin 2 r cos 2 ξ dφ 2 1 + sin 2 r sin 2 ξ dφ 2 2 + cos 2 r dφ 2 3 , We consider a NATD associated with two Killing vectors, The gauge symmetry can be fixed as z(σ) = 1 and x 1 (σ) = 0 , 7 The operator d is useful also in GSE. In GSE, the R-R fields have the dual-coordinate dependence as A = e −I mx mĀ (x m ) and F = e −I mx mF (x m ) , and the relation F = dA reproducesF = e I mx m dA = dĀ−ιIĀ . By considering (Ā,F ) as the dynamical fields, we obtain the relation in GSE (A.10). See [148] for more detail. and the AdS parts of the transformation matrices are For simplicity, we denote (u µ ) ≡ (x 0 ,x 1 , x 2 , x 3 ) , and then the dual background becomes Regarding the R-R fields, the first GL(D) transformation does not change the Page form and the next B-transformation gives The Abelian T -dualities along z and x 1 directions give From this Page form, we find Then, by introducing I = f ba b∂a =∂ z , they satisfy the type IIB GSE.
In order to obtain a solution of the usual supergravity, we perform a formal T -duality along the z-direction. By using the T -duality rule (A.14), we obtain a simple type IIA solution, (5.26)

AdS 3 × S 3 × T 4 with NS-NS and R-R fluxes
In order to demonstrate the effectiveness of our formula, let us consider a more non-trivial example. We consider the AdS 3 × S 3 × T 4 with the NS-NS and the R-R fluxes, where p and q are constants satisfying p 2 + q 2 = 1 . The Page form has the following form: Then, we consider the generalized Killing vectors, and the dual componentṽ 2 is important. Here, we choose the gauge as t(σ) = 1 and x(σ) = 1 .
The AdS parts of the transformation matrices are 30) and the NS-NS fields and the Killing vector take the form For the R-R fields, the first GL(D) transformation makes the replacement in the Page form (5.28), and by further acting e Λ f ∧ and T t · T x , we obtain the Page form in the dual background, where They satisfy the GSE under the original constraint p 2 + q 2 = 1 .
By performing a formal T -duality alongt-direction, we obtain which is a solution of type IIA supergravity.
We consider the following three Killing vectors, that satisfy the algebra The (x 1 , x 2 , x 3 ) parts of the matrices are 39) and the gauge symmetry is fixed as x i (σ) = 0 (i = 1, 2, 3) . The dual background becomes and this is a solution of type IIA GSE.
Again, by performing a formal T -duality along thex 1 -direction we obtain a solution of type IIB supergravity, We note that, as discussed in [147], some supergravity solutions, obtained by a combination of NATD and a formal T -duality, can be also obtained from another route, a combination of diffeomorphisms and the Abelian T -dualities. Similarly, solutions obtained in this section may also be realized from such procedure.

Poisson-Lie T -duality/plurality
Here we study a more general class of T -duality, known as the Poisson-Lie T -duality [37,38] or T -plurality [55]. We can perform the PL T -duality/plurality if the target space has a set of vectors v a satisfying the dualizability conditions [37] [ The non-Abelian T -duality (withṽ a = 0) can be regarded a special casef bc a = 0 . We begin

Review of PL T -duality
We review the PL T -duality as a symmetry of the classical equations of motion of the string sigma model. To make the discussion transparent, we first ignore spectator fields y µ (σ), which are invariant under the PL T -duality. As studied in [37,38], it is straightforward to introduce the spectators, and their treatments are discussed in section 6.2.4.
Let us consider a sigma model with a target space M , on which a group G acts transitively and freely (i.e. M itself can be regarded as a group manifold), Under an infinitesimal right-action of a group G , the coordinates x m are shifted as where T a (a = 1, . . . , n) are the generators of the algebra g satisfying and v m a are the left-invariant vector fields satisfying In general, the variation of the action becomes If v m a satisfy the Killing equation £ va E mn = 0 , equations of motion for x m can be written as dJ a = 0 . The PL T -duality is a generalization of this duality when the vector fields v a satisfy In this case, the variation becomes It is important to note that the condition (6.10) and the identity show a relation By considering the vector spaceg as the dual space of g , T a ,T b = δ b a , the relation gives the structure of the Lie bialgebra. By defining an ad-invariant bilinear form as the commutation relations on a direct sum d ≡ g ⊕g are determined as 18) and the pair of two algebras can be regarded as that of the Drinfel'd double D . Given the structure of the Drinfel'd double, the differential equation (6.10) can be integrated as [37,38] where the matrices a and b are defined by andÊ ab is a constant matrix (that corresponds to E ab (x) at g = 1). We can check that the E mn given by (6.19) indeed satisfies (6.10). 8 Now, we rewrite the relation (6.13) c that can be derived from (6.20) (see [44]).
into two equivalent expressions (by following the standard trick [10] in the Abelian case), They can be neatly expressed as a self-duality relation, where indices A, B, · · · are raised or lowered with η AB and its inverse η AB . In terms of the metric H AB , the relation (6.19) can be expressed as whereĝ ab ≡Ê (ab) ,B ab ≡Ê [ab] , and we can easily obtain where we have used 9 P(σ) ≡ dl l −1 = g ℓ a T a +r aT a g −1 = P B (Ad g ) B A T A . the equations of motion can be expressed aŝ  30) and the matrix U is defined as For later convenience, we rewrite the twist matrix as where we have defined and used r a = (a − ⊺ ) a b ℓ b . Then, in terms of E mn (x) , (6.30) can be expressed as 35) and similarly, the dual background is In a special case, wheref ab c = 0 , by parameterizingg = ex aT a , we obtainr = dx aT a , Π ab = 0 , andΠ ab = −f ab cx c . This is precisely the dual background for NATD. In the dualized background, in general, the isometries are broken, and in the traditional NATD, we cannot recover the original model. However, the background has the form whereẽ a =ṽ a =∂ a , and we find that the dual background is T -dualizable, £ṽa E mn =∂ a E mn = f bc aṽbmṽcn .  [161,[166][167][168][169][170]. The correspondent of the DSM for the PL T -duality is studied in [40,41,64,[171][172][173], and that approach will be useful to manifest the PL T -duality.

PL T -plurality
The Lie algebra d of the Drinfel'd double D can be constructed as a direct sum of two algebras g andg , which are maximally isotropic with respect to the bilinear form ·, · , and the pair (d, g,g) is called the Manin triple. In general, a Drinfel'd double has several decompositions into the Manin triples, and this leads to a notion of the PL T -plurality [55]. More concretely, let us consider a redefinition of the generators T A of d , such that the new generators also satisfy the algebra of the Drinfel'd double, 40) and the bilinear form is preserved, The latter condition shows that the matrix C A B should be a certain O(n, n) matrix. Since the rescaling of the generators is trivial, we choose C A B as an SO(n, n) matrix.
The transformation of the background fields under the SO(n, n) transformation can be found in the same manner as the PL T -duality. Starting with a background E ′ mn satisfying again we obtain the same equations of motion From the identification, l = l ′ , we obtain 44) and the relation between the untwisted fields becomeŝ The generalized metric in the transformed frame also has the form, the relation between the original and the dual generalized metric is In terms of E mn (x) , the original background is while the dual background is

Duality rule for dilaton
The transformation rule for the dilaton was studied in [64] in the context of the PL T -duality.
This was improved in [55] in the study of the PL T -plurality. In our convention, the result is whereΦ(x) is an arbitrary function. By using the formula (3.28), we obtain Namely, the duality rule for the DFT dilaton is Ifd (or equivalentlyΦ) is constant, this duality rule coincides with the recent proposal [138], where the PL T -duality was studied by utilizing "the DFT on a Drinfel'd double" proposed in [137]. There, it was shown that the dilaton transformation rule is consistent also with [140].
Moreover, when the dual algebra is Abelianf ab c = 0 , we have |det(v ′m a )| = 1 and the result (3.30) known in NATD is also reproduced as a particular case.
In fact, as demonstrated in [55], the PL T -plurality works even ifd has a coordinate dependence. A problem is that when e −2d depends on the original coordinates x m , it is not clear how to understand the x m -dependence in the dual model. A prescription proposed in [55] is as follows. We first identify the relation between coordinates x I = (x i ,x i ) and x ′I = (x ′i ,x ′ i ) through the identification, (6.56) We next plug the relation Then, the relation (6.55) should be understood as

Covariance of equations of motion
In the approach of [137,138], the PL T -duality was realized as a manifest symmetry of DFT.
We here discuss the covariance under a more general PL T -plurality by using the gauged DFT.
The approach may be slightly different from [137,138] but essence will be the same.
In PL T -dualizable backgrounds, the generalized metric always has a simple form Since the twist matrix U is explicitly determined, we can compute the generalized fluxes F ABC and F A defined in (2.20). In fact, as shown in [138], in PL T -dualizable backgrounds, the three-index flux is precisely the structure constant of the Drinfel'd double,  [44] for useful identities).
We can also compute the single-index flux as By using the expression for the DFT dilaton (6.54), e −2 d = e −2d |det(r a m )| |det a| , we find where we have used a b e a c f f fe a = f cb e a e a and ∂ m a a b = a a c f cd b ℓ d m . As we discuss below, for the covariance of the equation of motion under the PL T -plurality, F A needs to transform covariantly. However, for example, in a particular cased = 0 , we find that F A dose not transform covariantly. Indeed, we have F A = 0 in the duality frame wherẽ f ab a = 0 while F A appears in the frame wheref ab a = 0 . Therefore, in order to transform F A covariantly, we eliminate the non-covariant term by adding a vector field X M as which was suggested in [138]. This shift is a bit artificial, but without this procedure, we need to abandon all Manin triples with non-unimodular dual algebra. In fact, this shift is precisely the modification of DFT equations of motion (2.38), that reproduces the GSE after removing the dual-coordinate dependence. After this prescription, we obtain a simple flux Under the PL T -plurality T ′ A = C A B T B , the generalized fluxes are mapped as

65)
10 This is non-trivial, because in general the derivative DA does not transform covariantly D ′ A = CA B DB , which can check by performing a coordinate transformation x ′M = x ′M (x) through (6.56). Therefore, at the present time, the covariance of FA needs to be checked on a case-by-case basis. Of course, whend is constant, the covariance is manifest because FA = 0 and F ′ A = 0 .
by introducing X M when the dual algebra is non-unimodular. According to (6.45), the untwisted metricĤ AB is also related covariantly, Before moving on to the R-R sector, we make a brief comment on the vector field I m .
In order to reproduce the (generalized) supergravity from (modified) DFT, we need to choose the standard section∂ m = 0 . Therefore, ifd has a dual-coordinate dependence, we should make a field redefinition. Supposing thatd only has a linear dual-coordinate dependencē d =d 0 (x m ) + d mx m , we make a field redefinition.
Then, the dual-coordinate dependence disappears from the background. Note that this is different from the shift (6.63) but just a field redefinition. In the following, when display a (generalized) supergravity solution, we always make this redefinition.
Before studying the R-R fields, let us make a comment on the Killing vector I m . In the case of NATD, the Killing vector I m is given by (4.1), but apparently, (4.1) is different from the formula (6.63) by the factor 2 . Here, we will roughly sketch how to resolve the discrepancy by using the redefinition (6.67). In the case of NATD, ∂ m |det(v m a )| = 0 and ∂ m Φ = 0 are usually satisfied in the original background (under the gauge fixing x m = c m ). Then, we have where we used v m a = δ m a in the dual theory. In a general setup, (4.1) does not work correctly, and we use the results discussed in this section.

Duality rule for R-R fields
Now, let us determine the duality rule for the R-R fields. We will first find the duality rule from a heuristic approach, and then clarify the result in terms of the gauged DFT.
Then, the energy-momentum tensor made from the combination e d (h) F (h) is as expected E ′ M N . However, an important thing is that, the actual DFT dilaton is given by 76) and e d (h) is related to e d ′ as Therefore, if we identify the dual R-R polyform as The so-called Clifford vacuum |0 is defined by Γ a |0 = 0 . By using a nilpotent operator the Bianchi identity can be expressed as As it is well-known in the democratic formulation [156,159], the Bianchi identity is equivalent to the equations of motion when the self-duality relation G p = (−1) Now, we require the dualizability condition for the R-R fields as / ∂|F = 0 , (6.85) which will be the same as the proposal of [138]. Then, the Bianchi identity or the equation of motion for the R-R fields becomes an algebraic equation We call the objectF , the untwisted R-R fields, and once the untwisted R-R fieldsF ′ a 1 ···ap in the dual background is obtained from (6.79), the Page form in the dual background can be constructed as

Spectator fields
In the following, we consider more general cases where spectator fields are also included.
Namely, we suppose that the original model takes the form, In the following, we denote the coordinates as (x m ) = (y µ , x i ) (i = 1, . . . , n) . By assuming that the background field (E mn ) = Eµν E µb Eaν E ab satisfies the condition, This reduces to (6.48) when there is no spectator field. An important difference is thatÊ mn is not necessarily constant, but can depend on the spectator fields y µ ,Ê mn =Ê mn (y µ ) . The dependence should be determined from the DFT equations of motion and it is independent of the structure of the Drinfel'd double.
In terms of the generalized metric, we can clearly see that the above relation (6.91) is a straightforward generalization of (6.30), where (x M ) = (y µ , x i ,ỹ µ ,x i ) . The T -plurality transformation (6.45) is also generalized aŝ The dilaton will also have an additional dependence on the spectators similar to (2.29), We also suppose that the untwisted R-R fields are functions of the spectatorsF =F(y µ ) .
Then, by defining the fluxes F ABC and F A from U M A (x) and d(x) , we again obtain (6.96) Here, again we need to perform a shift ∂ M d → ∂ M d + X M (6.63) when the dual algebra is non-unimodular.
The requirement (2.30) is automatically satisfied with our twist matrix, and by using (2.31), the dilaton equation of motion becomeŝ we can easily see that D CĤAB = ∂ CĤAB and D Ad = ∂ Ad are also transformed covariantly,  In this section, we show an example of the Poisson-Lie T -plurality. As already mentioned, the Lie algebra d of the Drinfel'd doubles can be realized as a direct sum of two maximally isotropic algebras g andg and the pair (d, g,g) is called the Manin triple. By following [54], we denote the pair simply as (g|g) . The classification of six-dimensional real Drinfel'd doubles was worked out in [54], and there, the following series of Manin triples, which corresponds to a single Drinfel'd double d , has been found: Here, the characters in each slot denote the Bianchi type of the three-dimensional Lie algebra, Manin triples was studied in [55]. In [55], since the initial background is the flat space (or the Bianchi type V universe) the R-R fields were absent in any of the dual backgrounds. Moreover, due to a problem in the treatment of the dual-coordinate dependence of the dilaton, only the three backgrounds were discussed (see also [56][57][58] for the dilaton puzzle), In this section, we identify the AdS 5 × S 5 solution as a background with (5|1) symmetry, and write down all of the eight backgrounds with the Manin triples given in (7.1).
For convenience, we summarize the procedure of the PL T -plurality.
Untwisted fields Ĥ AB (y),d(y),F(y) We first prepare the untwisted fields {Ĥ AB (y µ ),d(y µ ),F(y µ )} that satisfy By using the generators T A in each frame, we construct the twist matrix U as Then, by twisting the untwisted fields, we construct the DFT fields as The functiond(x) is given in the initial configuration, and after the PL T -plurality, it is rewritten in the new coordinates determined by the invariance of l = g(x i )g(x i ) . When d(x) has a linear dual-coordinate dependence d ix i , we make a redefinition and absorb the dependence into the Killing vector, We start with the AdS 5 × S 5 background, in a non-standard coordinate system, where ds 2 S 5 ≡ dr 2 + sin 2 r dξ 2 + cos 2 ξ sin 2 r dφ 2 1 + sin 2 r sin 2 ξ dφ 2 2 + cos 2 r dφ 2 3 , ω 5 ≡ sin 3 r cos r sin ξ cos ξ dr ∧ dξ ∧ dφ 1 ∧ dφ 2 ∧ dφ 3 .
Note that if we choose the untwisted fields as the purely NS-NS solutions studied in [55] can be recovered.

(1|5): type IIA GSE
In order to consider the NATD background, we perform a redefinition of generators, and give a parameterization, Then, from the identification with the original background, we find the following relation between the coordinates: From this relation, we can identifyd as For notational simplicity, in the following we drop the prime.
The untwisted fields in this frame become , sin 2 r, sin 2 r cos 2 ξ, sin 2 r sin 2 ξ, cos 2 r , and we twist them by using the quantities, The resulting metric and the B-field are Since the dual algebra 5 is non-unimodular, we need to introduce the Killing vector We can check that the flux F A is transformed covariantly from the original one F which shows that the equations of motion are transformed covariantly. In order to make the background as a solution of GSE, we make the redefinition (6.67) which gives After this redefinition, the dual geometry becomes which is a solution of type IIA GSE. We can explicitly check that this background has the (1|5) symmetry, A formal T -duality along the x 1 -direction gives a simple solution of type IIB supergravity We next perform the following redefinition from the original (5|1) generators: This time, we provide a parameterization (7.36) and the coordinates are related to the original ones as Then, in this frame,d becomes Again we remove the prime, and then the (t, x 1 , x 2 , x 3 , z)-part of the untwisted metric becomes In order to obtain the untwisted R-R fields, it may be useful to decompose the matrix C into Then, the T -duality along the x 2 -direction and the GL(3) transformation givê In order to obtain the twist matrix, we compute Again, the flux F A is transformed covariantly, The background fields are determined as 45) and this is a solution of type IIA supergravity.

(1|6 0 ): type IIB GSE
The NATD of the (6 0 |1) background, namely (1|6 0 ) can be realized by We give a parameterization In order to determined , it is enough to identify the coordinate x 1 , and we find Note that the appearance of the dual-coordinate dependence was discussed in [55], but at that time, DFT had not been developed and the interpretation was not clear.
We can construct the twist matrix U from Although the dual algebra is unimodular, in order to absorb the dual coordinate dependence ind , we make a field redefinition (6.67) and obtain e −2d = 1 , After the redefinition, we obtain a solution of type IIB GSE, It is important to note that the duality (6 0 |1) → (1|6 0 ) is a NATD for traceless structure constants. In the literature, it has been discussed that if the structure constants are traceless, the NATD background satisfies the supergravity equations of motion, and it is important to clarify the consistency with the example discussed here. Of course, the existence of the R-R fields is not important here. As already mentioned, we can obtain a purely NS-NS solution, by starting with the untwisted fields (7.19). The (6 0 |1) background is while its NATD, namely the (1|6 0 ) background, is a GSE solution, (7.55) It will be interesting to study string theory on these backgrounds in detail.
We also note that, in the (1|6 0 ) background (7.53), if we perform a formal T -duality along the x 3 -direction, we obtain a solution of type IIA supergravity, (7.56)

(5|2.i): type IIB SUGRA
In order to obtain the Manin triple (5|2.i), we perform a redefinition Again we consider a parameterization, and from the coordinate transformation, we obtain e −2d = e −2 x 1 = e 2 x ′1 . (7.59) The necessary quantities are obtained as and this is a solution of type IIB supergravity.
7.6 (2.i|5): type IIA SUGRA We next consider a transformation, and provide a parameterization, The coordinate transformation gives Again, we compute and we can check that the flux is covariantly transformed, Since the dual algebra 5 is non-unimodular, we have We thus expect that this background is a solution of the GSE. However, according to the field redefinition (6.67), we obtain e −2d = 1 , As the result, we obtain a solution of the conventional type IIA supergravity Namely, even if the dual algebra is non-unimodular, the background can satisfy the usual supergravity equations of motion. This is a remarkable example of such unusual cases. 7.7 (5.ii|6 0 ): type IIB GSE We next consider and give a parameterization, We then obtaind as From a straightforward computation, we obtain the twist matrix U , and the flux is covariantly transformed Since the dual algebra is unimodular, originally we have I m = 0 . However, due to the dual-coordinate dependence ofd , we make the field redefinition (6.67) and obtain After the redefinition, we obtain a solution of type IIB GSE, which is defined on the region ∆ 2 ≡ 4 t 4 (e 2 x 2 +1) z 2 + e 2 x 2 z 6 (2 − 2 e x 1 + e x 2 ) 2 ≥ 0 . (7.81) A formal T -duality along the x 3 -direction gives a solution of type IIA supergravity, 7.8 (6 0 |5.ii): type IIA SUGRA Finally, we consider a redefinition, This time, we consider a parameterization 13 which leads to By using Since the dual algebra 5.ii is non-unimodular, the Killing vector becomes Then, after the redefinition, we obtain a solution of type IIA supergravity, , (7.90)

Summary of results
We discussed two approaches to the non-Abelian T -duality. One is the traditional NATD, obtained by integrating out the gauge fields associated with non-Abelian isometries, and the other is the PL T -duality/plurality, which is based on the Drinfel'd double.
In NATD, a closed-form expression for the duality rules including the R-R fields was known only for a certain isometry group SU(2) , but we proposed a general formula by assuming that the isometry group freely acts on the target space.
Three of them are solutions of GSE. There are two origins of GSE; one is the Killing vector ba b v m a that appears when the dual algebra is non-unimodular, and the other is the dual-coordinate dependence ind . In the last two examples, the two contributions are canceled with each other, and they are solutions of the usual supergravity even though their dual algebras are non-unimodular. In the literature, whend has a dual-coordinate dependence, since its interpretation is not clear in string theory or supergravity, such Manin triple was ignored. However, in DFT, we can treat the dual coordinates in the same ways as the physical coordinates, and we can lift the restriction. In this way, the PL T -plurality is a solution generating technique of the DFT, rather than the usual supergravity.

Discussion and outlook
As we discussed, if we consider a supergravity solution that contains a four-dimensional Minkowski spacetime, ds 2 = f 2 (y) η µν dx µ dx ν + · · · , we can choose the coordinates such that the (5|1) symmetry manifest. Then, as long as the B-field isometric along the three Killing vectors, we will obtain a family of eight solutions similar to the case of AdS 5 × S 5 .
Moreover, low-dimensional Drinfel'd doubles are already classified in [52][53][54] and a useful list is given in section 3 of [54]. If we have a DFT solution with an isometry algebra g , we may find a series of Manin triples, (g|1) ∼ = · · · ∼ = · · · , (8.1) and obtain a chain of DFT solutions. We may also start from a background with a (g|g)symmetry. For example, as discussed in [50], the Yang-Baxter deformed backgrounds are also PL T -dualizable. Indeed, a Yang-Baxter deformed background has the form In the traditional approach to NATD, we introduced the generalized Killing vector (V M a ) = (v m a ,ṽ am ) . When the dual componentsṽ am are present, we cannot regard the NATD as a particular case of the PL T -plurality. Also when the generalized Killing vectors depend on the spectator fields y µ , we cannot realize them as the left-invariant vector fields. In this sense, the traditional NATD is not completely contained in the PL T -plurality discussed here. It is interesting to study whether it is possible to generalize the PL T -plurality such that the traditional NATD can be realized as a particular case. In the realm of NATD that is going beyond the PL T -plurality, it is not ensured that the dual background is a solution of DFT. By the definition of the NATD, the duality rules for the metric and B-field should not be modified, but the transformation rule for the dilaton and I m may be modified from (5.2). Indeed, a modification was required in the example (6 0 |1) → (1|6 0 ) of the PL T -plurality. It will be an important task to determine the general rule for the dilaton and I m that is consistent with the DFT equations of motion. Once the modification of the rule for the dilaton is determined, the modification of the rule for the R-R fields (by an overall factor) can be also determined, and then we can check the equations of motion for the R-R fields.
In this paper, in both approaches, we have assumed that the isometry group acts on the target space freely, or without isotropy. If the assumption is not satisfied, we cannot take a gauge x i = c i and we need to consider a more non-trivial gauge fixing. Treatments in such cases are discussed for example in [19,73,77,174] for the NATD, and in [41,45,47] for the PL T -duality. It is an interesting future direction to check whether the DFT equations of motion are covariantly rotated even in such general cases.

Toward non-Abelian U-duality
Another important future direction is an investigation of the non-Abelian U -duality. As an attempt toward the non-Abelian U -duality, let us first consider an extension of the traditional NATD. As a natural extension of (3.3), let us consider the following setup [154], where F 4 ≡ dC 3 is the four-form field strength in the eleven-dimensional supergravity. We definev (1) ab ≡ ι v bv We also assume the existence of 1-forms ℓ a ≡ ℓ a i dx i , that are dual to v a (ι va ℓ b = δ b a ), and then we find that the action 14 bc . Here, by following the approach of [175] (see also [176]), we have introduced antisymmetric Lagrange multipliers y ab = −y ba that will ensure F a = 0 .
In the Abelian limit, we can realize v i a = δ i a and ℓ a = δ a i dx i , and then we can always choose a gauge x i = 0 . By further assumingv (2) a = −ι va C 3 , the action reduces to This is precisely the action discussed in [175,176] and (8.6) can be regarded as a natural extension. However, unlike the case of the string action, it is not clear how to eliminate the gauge fields A a , and at the present time, we do not know how to obtain the dual action.
A more promising approach may be the following approach based on a generalization of DFT. The U -dual version of DFT is known as the exceptional field theory (EFT) [176][177][178][179][180][181][182][183] and it is actively studied. In DFT, the generalized coordinates are (x M ) = (x m ,x m ) and the dual coordinatesx m are associated with the string winding number. On the other hand, in EFT, we introduce the dual coordinates for all of the wrapped branes that are connected by U -duality transformations. For example, in M-theory on a n-torus, we have the M2-brane, the M5-brane, and the Kaluza-Klein monopople, and more exotic branes in general, and correspondingly, we introduce the generalized coordinates as (x I ) = (x i , y i 1 i 2 , y i 1 ···i 5 , y i 1 ···i 7 , i , · · · ) (i = 1, . . . , n) . In such extended space, the generalized metric M IJ has been constructed in [176,180], and it contains the bosonic fields, such as the metric g ij , the 3-form and 6-form potentials, C i 1 i 2 i 3 and C i 1 ···i 6 . It is a natural generalization of the generalized metric H M N in DFT.
In DFT, the section condition η IJ ∂ I ∂ J = 0 reduces the doubled space to the physical subspace. The section condition in EFT (for n ≤ 6) also has a similar form η IJ;K ∂ I ∂ J = 0 , where η IJ;K is known as the η-symbol and it has additional indexK transforming in another representation (see [184] for the explicit form of the η-symbol). When all of the fields depend only on the coordinates x i of (8.9), we find η IJ;K ∂ I ∂ J = η ij;K ∂ i ∂ j = 0 ∵ η ij;K = 0 , (8.11) and the section condition is satisfied. This n-dimensional solution is called the M-theory section. Another solution, called the type IIB section, was found in [185], and in order to discuss the type IIB section, it is convenient to reparameterize the coordinates as 15 (x M ) = (x m , y α m , y m 1 m 2 m 3 , y α m 1 ···m 5 , y m 1 ···m 6 ,m , · · · ) (m = 1, . . . , n − 1 , α = 1, 2) , (8.12) where the dual coordinates are associated with the type IIB branes. If the fields depend only on the x m , the section condition is again satisfied because η mn;P = 0 . Since we cannot introduce any more coordinate-dependence, the subspace spanned by x m is also a maximally isotropic subspace, although it is (n − 1)-dimensional unlike the M-theory section. In this way, a single EFT can be reproduced from the two viewpoints, M-theory and type IIB theory. 15 The explicit relation between x I and x M was determined in [186].
One of the key relation in the PL T -duality is the self-duality relation, η ABP B =Ĥ AB * P B ,P(σ) = dl l −1 .

(8.13)
This is a covariant rewriting of the string equations of motion, but a similar equation for the M2-or M5-brane theory has been discussed in [187,188] for the SL(5) and SO (5,5) case, and in [189] for higher exceptional groups. For the Mp-brane (p = 2, 5), it has a similar form η IJ ∧ P J = M IJ * P J , (8.14) where η IJ is some (p − 1)-form that contains dx i and the field strengths of the worldvolume gauge fields. In the case of the flat torus, the equations of motion give dP I = 0 and we find the on-shell expression P I = dx I . On the other hand, by requiring a certain "dualizability condition" on M IJ appropriately, the equations of motion may lead to P = dl l −1 , where l is an element of a certain large group E with dimension D . The corresponding algebra e will be endowed with a bilinear form, corresponding to the η-symbol. Then, the U -dual version of the PL T -plurality may be the equivalence between sigma models with n-or (n − 1)-dimensional target spaces that have an isometry algebra [T a , T b ] = f ab c T c satisfying η ab;Â = 0 . The identification of the detailed structure of the group E and the systematic construction of the twist matrix U , whose flux gives the structure constant of e, are interesting future directions.