DFT in supermanifold formulation and group manifold as background geometry

We develop the formulation of DFT on pre-QP-manifold. The consistency conditions like section condition and closure constraint are unified by a weak master equation. The Bianchi identities are also characterized by the pre-Bianchi identity. Then, the background metric and connections are formulated by using covariantized pre-QP-manifold. An application to the analysis of the DFT on group manifold is given.


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Since Siegel proposed to formulate spacetime duality in superspace [1,2] double field theory (DFT) has been developed and investigated by many authors [3][4][5][6][7][8] with the aim to construct an effective theory of string with manifest T-duality symmetry. The gauge transformation and diffeomorphism are unified into the generalized Lie derivative of doubled space [5] similarly as done in generalized geometry [9][10][11][12]. For previous work on generalized geometric structures in DFT, see for example [13] and references therein.
However, unlike generalized geometry, the gauge algebra of DFT generated by the generalized Lie derivative does not close. To achieve closure we need to constrain the algebra. One way to do so is to impose the section condition (strong constraint), a particular solution of which reduces the DFT action to the standard action of supergravity. On the other hand, the section condition is not the only possibility to close the DFT algebra [8,14]. This was demonstrated via a Scherk-Schwarz [15] type compactification of DFT to the double torus, now known as generalized Scherk-Schwarz (GSS) compactification [8,[16][17][18]. In the GSS compactification, the fields depend also on the internal space coordinate. The doubled coordinates of the internal space enter in a very particular way through the GSS ansatz. It has been shown that the resulting fluxes can be identified with the structure constants of the gaugings of gauged supergravity theories, giving a geometric interpretation to them all, including the non-geometric fluxes. This is very interesting, since in the compactified direction the section condition is not imposed, i.e., the GSS compactification gives an alternative solution to the closure constraint.
One aim of this paper is to provide an unified and geometric characterization of the closure constraint based on the supermanifold formulation. The supermanifold considered here is a non-negatively graded QP-manifold, i.e. a supermanifold with a graded Poisson structure (P-structure) and a nilpotent vector field (Q-structure), and its generalization called a pre-QP-manifold [19][20][21][22][23]. The usefulness of the graded manifold approach to DFT has been first pointed out by Deser and Stasheff [24] and has been applied to the generalized Lie derivative. After their work, the canonically transformed Hamiltonian functions of DFT were introduced and the T-duality transformation and Bianchi identities in DFT, including geometric and non-geometric fluxes, were formulated [20,23,25]. Here we elaborate the closure constraint for the DFT algebra from, what we call here, the weak master equation and also formulate the Bianchi identity on a pre-QP-manifold. We develop a O(D, D) covariant formulation of the canonical transformation and give a geometrical characterization of the Bianchi identities by introducing the pre-Bianchi identity on the doubled space. This yields a geometrical characterization of the Bianchi identities given earlier in [25].
As we shall see, using the graded manifold approach, it becomes possible to characterize the closure constraint without referring explicitly to local coordinates. Thus, it is natural to consider its application to the formulation of DFT on a non-trivial background. For this purpose, we first give the covariantization of the pre-QP-manifold. Using this formulation, we examine the closure constraint obtained by the weak master equation. and the corresponding Bianchi identities obtained from the pre-Bianchi identities. Then we give an application for DFT on the group manifold.

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This paper is organized as follows. In section 2, we introduce the definition of pre-QP-manifolds and reformulate the QP-structures of DFT in a O(D, D) covariant form. Then, we discuss the O(D, D) covariant form of the canonical transformation and work out the closure constraint. In section 3, we introduce the generalized vielbein of DFT via a canonical transformation, thus providing a unified picture for the generalized flux and giving a systematic formulation of the Bianchi identities. Then, we analyze the GSS compactification of DFT by O(D, D) covariant canonical transformation. In section 4, we investigate DFT on the group manifold using the pre-QP-structures. The last section is devoted to conclusions and discussion.

DFT on pre-QP-manifold
In this section, we first briefly recall the original formulation of DFT and the generalized Scherk-Schwarz compactification [8,16,17,26]. Then, we review the supermanifold formulation of DFT. The advantage of the supermanifold approach is that its QP-structure gives a concise characterization of the underlying algebra/algebroid structure.
After this preparation, we formulate the DFT on a pre-QP-manifold and elaborate the closure condition of the generalized Lie derivative and the meaning of the section condition in this wider structure. We shall see that the failure of the closure of the algebra can be understood as a failure of the classical master equation on the QP-manifold. However, unlike the original supermanifold approach [23,25] we do not make use of the section condition, since it is not the only possibility to achieve closure of the algebra [8,24,27]. From the pre-QP structure formulation we obtain new criteria for the closure of the algebra of the generalized Lie derivative. We will also see that the O(D, D) transformation is still realized as a canonical transformation. DFT was developed with the aim to construct a manifest T-duality invariant formulation of the effective theory of superstring, and thus has manifest O(D, D) symmetry. In order to realize this symmetry geometrically, the winding sector is included as a Fourier transform of extra dimensions and the theory is defined on a 2D-dimensional doubled space. We denote coordinates of this doubled space as XM = (X M , X M ) where the index M runs from 1 to D and hatted indices are used for the doubled indices, so thatM runs from 1 to 2D. The partial derivative ∂M = (∂ M , ∂ M ) is also doubled correspondingly.
The dynamical fields in DFT are the generalized metric HMN and the generalized dilaton d.

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and the 2D-dimensional indices are lowered and raised by ηMN and its inverse ηMN , respectively. The generalized metric can be parametrized by the D-dimensional metric g M N and the 2-form field b M N as We introduce the generalized vielbein EM With the generalized metric and generalized dilaton, the DFT action of the NS-NS sector [7] can be written as: In DFT the diffeomorphisms and gauge transformations in the D-dimensional theory are unified into the generalized Lie derivative [5], similarly as in generalized geometry [9,10], and the algebra of the generalized Lie derivative characterizes the symmetry of DFT. The above action is invariant under the gauge transformation generated by the generalized Lie derivative L. The generalized Lie derivative of an O(D, D) vector VM with weight w is defined as Here, Λ is a generalized vector corresponding to the gauge parameter depending on the 2D dimensional coordinates, The weight of both generalized metric HMN and generalized vielbein EÂM is 0, and the generalized Lie derivative acts on them as

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By construction, the generalized Lie derivative contains the usual gauge transformation of the B-field and the D-dimensional diffeomorphism: when we parametrize the generalized metric as (2.3) and take a special solution of the section condition, called the supergravity frame, by requiring that all fields are independent of the dual coordinateX M , the DFT action reduces to the NS-NS sector of the supergravity action. In the supergravity frame the generalized Lie derivative reduces to the D-dimensional diffeomorphism and the gauge transformation of the B-field: (2.11) where L λ is the standard Lie derivative with respect to a D-dimensional vector λ. The antisymmetrization of the generalized Lie derivative of DFT is known as C-bracket In the supergravity frame, the C-bracket reduces to the Courant bracket.
The obstruction for defining a generalization of the diffeomorphism by the generalized Lie derivative is that L Λ does not satisfy the Leibniz rule. That is does not vanish, which is apparently satisfied by the standard Lie derivative L λ . Vanishing of the above ∆ M (Λ 1 , Λ 2 , V ) means that the commutator of the generalized Lie derivative satisfies which is also called the closure condition. Closure is always guaranteed, when the section condition is imposed ηMN (∂M Φ)(∂N Ψ) = 0 (2.16) where Φ and Ψ denote any fields and gauge parameters of DFT. However, the section condition is not necessary to achieve closure for the algebra.

Generalized Scherk-Schwarz (GSS) compactification
In this paper, we also analyze the generalized Scherk-Schwarz (GSS) compactification of DFT where the 2D-dimensional DFT is reduced to (D − n)-dimensional gauged supergravity. The generalized Lie derivative of the 2D-dimensional DFT is twisted by the GSS ansatz and the constraint of its internal space is relaxed compared to the section condition [8,[16][17][18]. For the GSS compactification, it is convenient to start with the DFT action in flux formulation, which is equivalent to the DFT action in generalized metric formulation (2.6) up to terms which vanish under the section condition [16]: EĈN is the generalized Weitzenböck connection. Note that, if we parametrize the O(D, D) vielbein by the D-dimensional metric g M N and 2-form b M N , we find the H-flux and the geometric f -flux in FÂBĈ. Alternatively, if we take the parametrization by the fieldsg M N and the two-vector β M N we obtain the so-called non-geometric fluxes. This result provides a unified picture of fluxes discussed in [23].
As in the original Scherk-Schwarz compactification ansatz [15], the generalized Scherk-Schwarz (GSS) compactification of DFT splits the 2D-dimensional target space with coordinate X into 2d-dimensional external space with coordinate X and 2(D − d)-dimensional internal space with coordinate Y, as X = (X, Y). Then, the ansatz for the generalized vielbein EÂM (X), generalized dilaton d(X) and gauge parameter ΛM (X) are [16,26], ΛM (X) = ΛÎ (X)UÎM (Y). (2.23) where the fields with hat, EÂÎ (X), d(X) and ΛÎ (X) are the vielbein, dilaton and gauge parameter of the reduced theory, respectively. We use the charactersÎ,Ĵ,K,L andĤ for the indices of the reduced theory, and the corresponding O(D, D) metric ηÎĴ . The matrix UÎM (Y) and its inverse UÎM (Y) which give the GSS twist are elements of O(D, D) and d(Y) is a scalar. The twist matrices are assumed to satisfy for any field of the reduced theory, which means that the twist occurs only in the internal space, and preserves the Lorentz invariance in the external spacetime of the reduced theory. By substituting the above GSS ansatz (2.21) into the generalized fluxes (2.19) and (2.20), we obtain the twisted generalized fluxes: where ΩÎĴK = UÎM ∂M UĴN UKN . In the GSS compactification, these fluxes fÎĴK, fÎ are assumed to be constant in the same way as in the original Scherk-Schwarz compactification. Imposing this ansatz the resulting theory becomes independent of the internal coordinate and the DFT action is reduced to the effective action of the so-called gauged double field theory (GDFT) [17]. The gauge algebra of GDFT is inherited from the original DFT. The corresponding generalized Lie derivative of a generalized vectorV (X) in the reduced theory can be derived from the one of the original DFT by substituting the GSS ansatz as (2.31) Thus, the generalized Lie derivative L of GDFT is defined on the reduced fields and gauge parameter which depend only on the external spacetime coordinate X.
The algebra of L closes by the closure constraint for GDFT fields and the Jacobi identity of the structure constant fÎĴK, (2.32) Note that the condition for closure for the internal space is relaxed compared to the solution by using the section condition.

Supermanifold formulation of DFT
In this section, first we summarize briefly the supermanifold formulation of DFT given in the refs. [20,23,24]. Then, we define a pre-QP-manifold which is a generalization of QP-manifold that can describe the algebraic structure of DFT. On the pre-QP-manifold the closure condition is relaxed, which fits to the generalized Lie derivative and gives a new understanding of the section condition. The algebra on a QP-manifold is a kind of simplified graded algebra as used in the BVand BRST-formalism. See e.g. [19] and references therein. There are two structures on this supermanifold. One is called P-structure, which specifies a graded Poisson bracket and JHEP04(2019)002 thus defines the derivation and the vector field on the supermanifold. The other is called Q-structure, which is a nilpotent Hamiltonian vector field Q, an analogue of the BRST charge, the nilpotency of which is imposed by the classical master equation. It is known that a QP-manifold of degree 2 gives a concise definition of a Courant algebroid [28] and thus fits to the formulation of the generalized geometry [9,10].
The QP-manifold, however, is too strict to apply to DFT, as one can imagine from the relation between the double geometry and generalized geometry or, more concretely, from the relation between the C-bracket and Courant bracket.
We shall see that on a pre-QP-manifold we have more freedom to accommodate the DFT algebra, and we can obtain a new closure condition for the generalized Lie derivative, which gives another alternative to the section condition used in the original DFT.

Pre-QP-manifold and derived bracket
A P-manifold of degree n is a pair (M, ω), where M is a graded manifold with Z graded coordinates and ω is a graded symplectic form of degree n defining a P-structure. In our context, we always consider the case with non-negative n. The graded Poisson bracket {−, −} is calculated from the graded symplectic form ω as Here X f is a Hamiltonian vector field defined by ι X f ω = −df . Using graded Darboux coordinates (q a , p a ), the graded Poisson bracket has a local coordinate representation given by (2.34) The graded Poisson bracket is of degree −n and satisfies the following graded version of skew symmetricity, Leibniz rule and Jacobi identity, In the P-manifold we can define the following canonical transformation. Let α ∈ C ∞ (M) be a degree n function. Then, a canonical transformation generated by α is defined by the following exponential adjoint action, for any smooth function f, g on M.

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On the P-manifold, we can also define a Q-structure by specifying a degree n + 1 function Θ ∈ C ∞ (M). This function Θ defines a degree 1 graded vector field Q as If this vector is nilpotent, i.e., Q 2 = 0, it is called a homological vector field, and the corresponding Θ is called a Hamiltonian function or a homological function associated to Q. The condition Q 2 = 0 is equivalent to the classical master equation, A triple (M, ω, Θ), i.e., a P-manifold with a homological function Θ is called a QP-manifold. The algebra on the n = 2 QP-manifold is equivalent to a Courant algebroid and as we mentioned, it is too restrictive for DFT. The pre-QP-manifold which is applicable to DFT is obtained by relaxing the classical master equation. The pair (ω, Q) is called a pre-QP-structure and the triple (M, ω, Q) is called a pre-QP-manifold without requiring the nilpotency Q 2 = 0 of the vector field Q in (2.40). Consequently, the classical master equation does not vanish, Nevertheless, also in this case we call Θ the Hamiltonian function of the pre-QP-manifold.
As we shall see, in the pre-QP-manifold approach, the classical master equation is replaced by another condition, and the section condition is just one of the possible solutions to that condition.
Since the definition of a canonical transformation is independent of the Q-structure, the equation ( There is another important object in the supermanifold formulation, namely, the derived bracket. For any degree n + 1 function Θ on a P-manifold, the derived bracket is defined as a bilinear operation for f, g ∈ C ∞ (M): (2.44) The derived bracket [−, −] is not necessarily graded antisymmetric. Using the graded anti-symmetry (2.35) and the Jacobi identity (2.37) of the Poisson bracket, we obtain the following identity of the derived bracket for any f, g, h ∈ C ∞ (M),

Generalized Lie derivative on pre-QP-manifold
In the following we give the formulation of the generalized Lie derivative of DFT on the pre-QP-manifold. It gives a characterization of the closure of the gauge algebra using the weak master equation.
In order to construct the supermanifold formulation of DFT, we take a 2D dimensional

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The symplectic structure on M is which leads to the following graded Poisson brackets, In order to formulate DFT on the pre-QP-manifold, it is necessary to introduce the following coordinates (Q ′ , P ′ ) which we call the DFT basis: (2.55) Then, the push forward j ′ * and pullback j ′ * of a generalized vector field V = VM ∂M is defined as where VM P ′M ∈ C ∞ (M). We can also define the similar map for a 1-form on M . In the following, we identify the generalized vectors VM ∂M with VM P ′M and omit to write j ′ * and j ′ * for simplicity. The O(D, D) invariant inner product for the generalized vector fields V and V ′ can be defined by using the graded Poisson bracket of the corresponding functions on the supermanifold, In order to formulate the generalized Lie derivative of DFT by using the pre-QPmanifold, we take the following degree 3 function constructed by ΞM and P ′M , It is straightforward to show that the derived bracket using this Θ 0 gives the generalized Lie derivative on a generalized vector field V ,

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as well as the generalized anchor map for a function f Thus, the Θ 0 in (2.59) is a suitable Hamiltonian for the supermanifold formulation of DFT.
On the other hand, it is easy to see that the classical master equation of Θ 0 , (2.41) does not vanish: This means that for the algebra of the derived bracket with Θ 0 , i.e., the generalized Lie derivative defined on a pre-QP-manifold, the closure condition should be generalized as discussed in the previous subsection. There, we have shown that the condition for the Leibniz identity of the derived bracket on a pre-QP-manifold is given by the weak master equation, (2.47). In the DFT case, the weak master equation is Eq. (2.63) gives the closure constraint of the generalized Lie derivative as follows: the gauge parameter of generalized Lie derivative is a generalized vector, thus on the pre-QPmanifold, we require the weak master equation for the generalized vectors V 1 , V 2 , V 3 : This condition is equivalent to the closure constraint of the generalized Lie derivative in DFT:

Bianchi identity and GSS twist in pre-QP-formalism
In this section, first we analyze the canonical transformation on pre-QP-manifold. We show that the generalized vielbein can be introduced by a canonical transformation from the general tangent frame to the local flat frame. The definition of the generalized vielbein as an O(D, D) covariant canonical transformation provides a unified picture for the generalized flux and gives a systematic way to formulate the Bianchi identities. We also show that the twist operation in the GSS compactification of DFT can be analyzed by an O(D, D) covariant canonical transformation. We show that the canonical transformation of the Hamiltonian function of DFT gives the twisted Hamiltonian function which realizes the generalized Lie derivative of gauged DFT (GDFT).

Canonical transformation on supermanifold
To define the generalized vielbein and flux of DFT, we need to introduce a local flat frame. Correspondingly, we generalize the supermanifold to in the same way as the DFT basis, Q ′M and P ′M defined in (2.53). Here, ηÂB and its inverse Canonical transformations are generated by degree 2 functions on the n = 2 supermanifold. All possible degree 2 functions made from P ′ andP ′ are given by linear combinations of the following functions where the parameters AM A , tM P andtB A are functions of XM = (X M , X M ). The function A generates GL(2D) transformations in general, and t andt generate O(D, D) transformations in each frame. We can, in principle, also consider the degree 2 functions including Q ′ andQ ′ . However, we do not need them in the following discussions. We could also introduce the degree 2 function made from ΞM , but they do not generate the canonical transformations of the fiber directions.
We discuss the canonical transformations generated by each function in (3.3) in the following.

Canonical transformation by A and generalized flux
The canonical transformation generated by A is where K and L are matrices defined as KN M = ηMP AÂP ηÂBABN and LÂB = AÂN ηNP AĈP ηĈB, correspondingly. Since the coordinates P ′M andP ′Â correspond to the basis of the generalized tangent vector and the local flat frame, respectively, we can identify the matrix AM where ΩMNP = EÂM EBN EĈP ΩÂBĈ with ΩÂBĈ = EÂM ∂M EBN EĈN . In the following, we consider the transformation with parameter θ = π 2 . In this case, the transformation rules simplify as Then, the twisted Hamiltonian function becomes, where FÂBĈ and ΩÂBĈ are the generalized flux and generalized Weitzenböck connection introduced in (2.19). Thus, we have obtained the generalized flux and the Weitzenböck connection as a flux generated by a canonical transformation of Hamiltonian Θ 0 . We have concluded that the second term in (3.12) is in fact a generalized flux by comparing with the explicit form of the vielbein given by EM A in (2.19). This can be proven also directly using the representation of the generalized flux defined by the derived bracket as follows: We regard the generalized vielbein as a set of the generalized vectors EÂ = EÂM P ′M on the pre-QP manifold. The first line is the definition of the generalized flux by the derived bracket. After applying the canonical transformation, we obtain the last line, where the Poisson bracket withP ′Â ,P ′B ,P ′Ĉ picks up the coefficient ofP ′Â ,P ′B ,P ′Ĉ in the transformed Hamiltonian e π 2 δ E Θ 0 . Thus, we obtain the representation of the generalized flux in pre-QP-manifold as This explains why the generalized flux is generated by a canonical transformation of the Hamiltonian.
Finally, we consider the O(D, D) covariant canonical transformation generated by the functiont in (3.3). This transformation leads to the same formula as in (3.15) and (3.16) with P ′M replaced byP ′Â and T byT , respectively: We comment on the closure condition. The closure condition changes by twists as explained in section 2.2.1. Since twist functions here do not include ΞM , the closure condition is not changed on a function f (X) of degree 0. A D-dimensional physical subspace is not changed by a twist, however generalized connections are introduced by twists in the closure condition of a generalized Lie derivative on generalized vector fields. We discuss a concrete example in eq. (4.56) for a general case in section 4.

Generalized Bianchi identity on pre-QP-manifold
In this subsection, we propose a formulation of the Bianchi identity on the pre-QP-manifold and derive the DFT Bianchi identities satisfied by the generalized flux and Weitzenböck connection. On a QP-manifold, the Bianchi identity can be obtained from the classical master equation, {Θ F , Θ F } = 0 for the Hamiltonian function Θ F twisted by a flux as discussed in [23]. Especially, the n = 2 QP-manifold can be applied to the generalized geometry and the H- For this purpose we introduce the most general Hamiltonian function Θ F which includes all possible fluxes. Since the DFT on pre-QP-manifold can be formulated by using only P ′ andP ′ , we need to consider the degree 3 Hamiltonian function consisting of (XM , ΞM , P ′M ,P ′Ĉ ). It can be written by using six arbitrary tensors on M =M × M , denoted byρ, ρ, F , Φ, ∆, Ψ: We have seen above that we can generate some of the fluxes by canonical transformation of the Hamiltonian Θ 0 (2.59) without flux. We show that the Bianchi identities for the corresponding fluxes can be obtained by using the two Hamiltonians Θ 0 , Θ F combined with a canonical transformation as follows: the Bianchi identity on pre-QP-manifold can be defined by introducing the following degree 4 function B ∈ C ∞ (M), where α is a canonical transformation function of degree 2.
Then, the condition on the pre-QP-manifold for the Bianchi identity is the vanishing of the function B, which we call pre-Bianchi identity: Since for the QP-manifold case the classical master equation is satisfied, the second term of B(Θ F , Θ 0 , e δα ) vanishes and the pre-Bianchi identity reduces to the classical master equation {Θ F , Θ F } = 0, which gives the Bianchi identities for the fluxes introduced in Θ F .
The generalized Bianchi identity of DFT can be given by the pre-Bianchi identity (3.23) in the following way. Here, we take the Hamiltonian function: where fields E, Φ and F are considered as independent objects. We take Θ  [8,27]. See also [29]. The fourth equation (3.29) gives another generalized Bianchi identity for ΦÂMN . The equation (3.30) does not give a new condition. Note that in the above derivation, we have used the Θ F given in (3.24) for simplicity. However, in principle, we can use the most general Hamiltonian with fluxes given in (3.21). As a result of the pre-Bianchi identity, we obtain the condition that the redundant fluxes vanish.

GSS twist as canonical transformation
In this section, we show that the GSS ansatz (2.21) can be understood as a canonical transformation. The generalized Lie derivative (2.31) and generalized flux (2.25) after the compactification will be derived by using the canonical transformations on the pre-QPmanifold.
For the GSS compactification in the supermanifold formalism, we also split the base manifold into internal and external spaces. We use the same notation for coordinates X = (X, Y) as in section 2.1.2, that is X is used for the 2(D − d)-dimensional external space and Y is used for the 2d-dimensional internal space.

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Furthermore, we introduce 2D-dimensional intermediate coordinates of degree 1 denoted by ( QÎ , PÎ ) whereÎ,Ĵ,K,L,Ĥ = 1, ..., 2D and we define the corresponding DFT basis as in the same way as Q ′M and P ′M introduced before. The constant matrix ηÎĴ and its inverse ηÎĴ are O(D, D) invariant metrics. The Poisson brackets for Q ′Î and P ′ I are The intermediate coordinate P ′ I of the supermanifold is associated to the basis of the generalized vector V (X) of the reduced theory with injection j ′ : We identify the generalized vectors on the reduced theory VÎ (X)∂Î with VÎ (X) P ′ I and again omit to write the maps j ′ * and j ′ * for simplicity.

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The canonical transformation generated by t in (3.3) is the O(D, D) transformation of frame PÎ . Using this degrees of freedom, we can define a more general form of GSS twists: When we assume that TÎĴ depends only on Y, we can regard the matrix TÎĴ UĴM (Y) as GSS twist matrix. In (3.37) and (3.38), the GSS twist is generated by the canonical transformation e π 2 δ U . On the other hand, when we take UÎM = δÎM in (3.39) and (3.40), the GSS twist is generated by the canonical transformation e δ T . In the following, we discuss the GSS twist by the canonical transformation e π 2 δ U . The twisted Hamiltonian function is given by the same equation as (3.12) where EÂM is replaced by UÎM (Y): Here, the Weitzenböck connection ΩÎĴK = UÎM ∂M UĴN UKN is made from UÎM (Y) and the resulting flux fÎĴK = 3 Ω [ÎĴK] is assumed to be constant by the GSS ansatz. The generalized Lie derivative on the reduced theory is derived from the parent theory as The right hand side is calculated using the property of the canonical transformation as From this we can read off that leads the following conditions for the generalized vectors and structure constants: Since the generalized vectors depend only on the external space coordinate X, the first condition is just the closure constraint for the external space. The second condition is satisfied by the Jacobi identity for the constant flux fÎĴK.
By the canonical transformation given by the dynamical generalized vielbein E, the Hamiltonian function Θ GSS is transformed as where FÂBĈ and fÎĴK are defined in (2.27) and (2.29), respectively.
The dynamical field in the reduced effective theory is in EÂÎ . Therefore, the generalized flux of the theory after the GSS compactification is calculated in superspace formalism by applying the canonical transformation as: Thus, we can obtain the representation of the generalized flux as This shows that the GSS twisted flux appears in the twisted Hamiltonian function in the same way as in (3.14). We summarize the correspondence between the DFT and the GSS compactified DFT objects on the pre-QP-manifolds in table 1. The Q-structure Θ 0 is replaced by Θ GSS in the GSS compactified DFT. Thus, the deformation of the background of DFT on the pre-QP-manifold is realized by the deformation of the Hamiltonian function.

Covariantized pre-QP-manifold and DFT on group manifold
We have seen in previous section that in GSS compactification, the information of the twisted background is confined to the Hamiltonian function Θ GSS in (3.41). The generalized Lie derivative and flux are then obtained simply by replacing the Hamiltonian Θ 0 of original DFT with the Hamiltonian Θ GSS twisted by the GSS ansatz. We consider here to replace the GSS background with a general non-trivial background. Therefore, we use again the intermediate O(D, D) frame as in GSS case, which is denoted by the indicesÎ,Ĵ · · · together with the general tangent frame and the flat local frame with indicesM ,N · · · andÂ,B, · · · , respectively. For notations, see appendix A. Then, we take the ansatz analogous to (2.21) for the total vielbein as EÂM = EÂÎ EÎM , where EÎM is the background vielbein and EÂÎ is the fluctuation vielbein. The geometry of the background is contained in the background vielbein and the dynamical fields are in the fluctuation vielbein.
Here, we formulate the double geometry on a non-trivial background with pre-QPmanifold. As we have seen in the previous sections, the structure of the generalized Lie derivative and the consistency conditions are characterized by the pre-QP-structure. For this purpose, we introduce the covariantized pre-QP-manifold and analyze the double geometry of the background. The background vielbein is an element of GL(2D) and the metric ηMN is not constant in general. As in the standard geometry, we introduce the generalized affine connection and the covariant derivative. The generalized Lie derivative and the fluxes are also considered in the background geometry. Then, the fluctuation is introduced on this background. We apply this formulation to DFT on the group manifold proposed in [30].
Here, RMNŜR andRMNÎĴ are curvature tensor defined by ΓMNP and WMÎĴ , respectively: where the covariant derivative of a tensor VÎM is defined as where EÎM is the inverse vielbein, EÎM EĴM = δ I J . Furthermore, we assume that φ acts trivially on the base manifold coordinate as φ(XM ) = XM . We also require the condition of Poisson map on the dual coordinate ΞN . Especially, The equation (4.32) gives the following condition on the spin connection: WMĴLηLK + WMKLηĴL = WMĴK + WMKĴ = 0. (4.33)

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In the present formulation, we also require the vielbein postulate (4.28), then the covariant derivative of ηMN in (4.31) automatically vanishes as a consequence of (4.32), (4.34) Now we are ready to introduce the DFT basis on the covariantized pre-QP-manifold as in the section 2.2.2. Explicitly, we have ηMN PN ), (4.35) where ηÂB is the O(D, D) metric on the local flat frame. The Poisson brackets among the DFT basis coordinates P ′M , P ′ I , Q ′M , Q ′Î are given by The canonical transformation φ acts on the DFT basis Q ′M , Q ′Î , P ′M and P ′ I as,

Pre-QP-structure and gauge algebra
Since the covariant coordinate Ξ ∇ M realizes the background covariant derivative, we can formulate the pre-Q-sturcture on the covariantized pre-QP-manifold. The pre-Q-structure written with Ξ ∇ M realizes the generalized Lie derivatives in background covariant form. The simplest Hamiltonian function is The derived bracket from this Hamiltonian function defines the covariant generalized Lie derivative L ∇ Λ with generalized vector Λ = ΛM (X)P ′M on a generalized vector V = VM (X)P ′M on the background where the covariant generalized Lie derivative is given by replacing the derivative in the generalized Lie derivative by ∇M : We define the canonically transformed Hamiltonian function using φ introduced in the previous subsection:  Here, we want to make some remarks. First, note that the vielbein postulate guarantees the equivalence between the generalized Lie derivatives (4.49) and (4.52) defined on the two different frames. Second, once the covariant generalized Lie derivative is defined, we can formulate the generalized torsion in DFT on the pre-QP-manifold. The generalized torsion of the background is then where TPNM = ΓPNM − ΓNPM + ΓMPN (4.54) Finally, note that we can consider other possibilities of Hamiltonian functions written with Q ′ and Q ′ , but in our discussion here for the DFT case it is sufficient to consider the generalized vectors in the P ′ and P ′ sector of the DFT basis.

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The closure condition of the generalized Lie derivative (4.52) is the weak master equation (2.47) for generalized vectors: The above weak master equation leads to the following condition for the spin connection WMÎĴ and arbitrary generalized vectors V 1 , V 2 and V 3 , (4.56) We discuss this condition order by order in the differential on generalized vectors. Then, the following conditions are sufficient to satisfy (4.56).
The first condition is the closure condition of the generalized Lie derivative and it is satisfied with the section condition. The second and third condition can be solved for various cases.
Here we just show that the solutions for ordinary DFT and DFT WZW are included.
2) The second condition (4.58) can be satisfied by Note that AÎM , AÂĴ and AÂM are GL(2D) matrices and have no correspondence to vielbeine EÎM , EÂÎ and EÂM in general. The transformation rules for P ′ , P ′ and Ξ ∇ M are, where the matrices κ and λ are defined as κMN = AÎM AĴN and λÎĴ = AÎN AĴN , respectively. By identifying A with the vielbein, AÎM = EÎM , κ and λ become δMN and δÎĴ , respectively. In this case, the canonical transformation for A = θE are written as e θδ E P ′M = P ′M cos θ + EÎM P ′ I sin θ, (4.67) e θδ E P ′ I = −EÎM P ′M sin θ + P ′ I cos θ. Then, the canonical transformation e θδ E of the Hamiltonian function Θ ∇ 0 is Applying the similar discussion to A, we can introduce the fluctuation vielbein EÂÎ .
When we take AÂÎ = π 2 EÂÎ , we obtain the canonical transformation rules as follows.  . The generalized flux can be calculated by the derived bracket similarly as (3.14), as (4.78)

Pre-Bianchi identities
Now we can consider the pre-Bianchi identity for DFT on covariantized pre-QP-manifold.
To define the B function (3.22), we take the Hamiltonian function with general fluxes as, and for Θ 0 , we take Θ ∇ 0 given in (4.50). Since, as we have seen before, the canonical transformations e π 2 δ E e π 2 δ E generate the fluctuation on the background Hamiltonian Θ ∇ 0 ,

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we choose the α in eq.(3.22) as π 2 E. Then, the B function is given by

Application to DFT WZW
In this section, we apply our discussions to DFT WZW and specify the pre-QP-structure. We assume the background space as a group manifold G, so we can regard the coordinate P ′ I of its tangent space T G as the generator of the Lie algebra of G by the injection map JHEP04(2019)002 j ′ * ( P ′ I ) = TÎ . Then, the derived bracket of P ′ I should reproduce the Lie bracket: The left hand side is calculated as, The pre-Bianchi identity is of the same form and we obtain (4.81)-(4.85) where the spin connection satisfies (4.90).

Summary and conclusion
In this paper we formulated the algebraic structure of DFT on a pre-QP-manifold in an O(D, D) covariant form and gave the closure condition of the corresponding generalized Lie derivative. On the pre-QP-manifold we have more freedom to accommodate the DFT JHEP04(2019)002 algebra: the classical master equation is relaxed to a weaker derived bracket, i.e., the weak master equation, which defines the closure condition for the generalized Lie derivative. The weak master equation is in fact the condition of the Leibniz identity on the pre-QPmanifold.
Introducing a local flat frame, we defined the generalized vielbein and investigated the canonical transformations and the generalized flux on the pre-QP-manifold. The definition of the generalized vielbein as an O(D, D) covariant canonical transformation provides a picture which unifies all fluxes to a single generalized flux and gives a systematic way to formulate the Bianchi identities: the generalized Bianchi identities for all possible fluxes can be read off from a single pre-Bianchi identity.
We have also shown that the GSS compactification fits to this formalism. To formulate the GSS compactification we introduce the intermediate coordinate. With this coordinate we can introduce the GSS twist by a canonical transformation intertwining between the general coordinate frame and the intermediate frame. The fluctuation vielbein is then obtained as the canonical transformation from the intermediate frame to the local flat frame. This splitting gives the right description of the generalized flux in gauged DFT (GDFT).
One advantage of the superfield formulation is its background independence as can be seen, e.g., from the special structure of the derived bracket. It shows that all information on the background is completely contained in the Hamiltonian function Θ GSS of the intermediate frame. From the geometrical point of view, it is natural to formulate the geometry by using connection and covariant derivative. Therefore, in the last section, we developed the covariantized pre-QP-manifold to formulate the background geometry and gave a consistent theory with the Hamiltonian Θ ∇ instead of Θ GSS .
One important observation is the algebraic property of the Ξ ∇ coordinate. It shows that the Poisson structure is preserved in original P Q coordinates as well as in the primed DFT basis. Note that the coordinate Ξ ∇ is fixed by the requirement of conservation of the P-structure and the vielbein postulate.
We have shown that the familiar geometric objects are obtained from the pre-QPmanifold through certain identifications. Thus, we have also shown the application of the superfield formulation to the group manifold case. A construction of DFT on the group manifold has been intensively discussed in [33] in the wider context of Poisson-Lie T-duality, which contains both abelian and non-abelian T-duality as special cases. The solution of the weak master equation in section 4 reduces consistently to the DFT WZW theory discussed in [33].
Finally, we discuss the relation of our approach to GSS compactification and DFT on group manifolds in supermanifold formulation. In section 3.3 we derived the GSS twist in terms of a canonical transformation. There, a GSS twist matrix UMN (Y) is introduced via the canonical transformation of the degree 2 function A = π 2 U in (3.35). The GSS twisted vielbein is (3.37) and the twisted Hamiltonian function is (3.41).
On the other hand, the GSS compactified DFT can be regarded as a covariantized DFT whose background manifold is a generalized twisted torus. From this point of view, the background vielbein EÎM is identified with the GSS twist matrix UÎM (Y). With this JHEP04(2019)002 identification, the total vielbein EÂM is identified with the GSS twisted vielbein. In our approach, this corresponds to the GSS twist matrix introduced by the canonical transformation e δ A in section 4.1.4. We identify the transformation function AÎM with the GSS twist matrix EÎM (Y). Then, the GSS twisted vielbein is obtained as In this case, the GSS twisted Hamiltonian function (4.50) becomes, The difference between the Hamiltonean function (3.41) and the one given above (5.2) is due to the fact that the former one was not written in a covariant form, while the latter one is covariant. Explicitly, we have However, these terms do not affect the generalized Lie derivative and we obtain the same formula for the generalized flux from the connection in covariantized formulation.

D Double Field Theory on group manifolds
Let us briefly recall the DFT on group manifold defined in ref. [30]. The DFT on group manifold has been developed as DFT WZW which is considered as a double field formalism of conformal string field theory (CSFT) for the Wess-Zumino-Witten (WZW) model [31,32].
From now on we take the background group manifold as the direct product of two Lie groups G = G × G and the O(D, D) metric ηÎĴ as

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We assume that the covariant generalized Lie derivative (D.18) reproduces the generalized Lie derivative of DFT WZW (D.24). Then, the spin connection is written by the structure constant as follows. WÎĴK = 1 3 FÎĴK.
(D. 25) In the following, we discuss the closure condition of the generalized Lie derivative (D.24) which takes the form Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.