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.


Introduction
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]. 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 [8,13]. This was demonstrated via a Scherk-Schwarz [14] type compactification of DFT to the double torus, now known as generalized Scherk-Schwarz (GSS) compactification [15,16,8,17]. 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-tion 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 [22,24] we do not make use of the section condition, since it is not the only possibility to achieve closure of the algebra [23,8,26]. 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. 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] 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 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 [15,16,8,17].
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 [15]: result provides a unified picture of fluxes discussed in [22].
As in the original Scherk-Schwarz compactification ansatz [14], 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 [25,15], The twist matrices are assumed to satisfy UÎM ∂M g(X) = ∂Î g(X) , (2.24) 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. 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) [16].
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 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, 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. [23,19,22]. Then, we define a pre-QP-manifold which is a generalization of QPmanifold 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. [18] and references therein. There are two structures on this supermanifold. One is called P-structure, which specifies a graded Poisson bracket and thus defines the derivation and the vector field on the supermanifold. The other is called Qstructure, 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 QPmanifold of degree 2 gives a concise definition of a Courant algebroid [27] 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 for f, g ∈ C ∞ (M), where |·| denotes the degree. 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.
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 corre- 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 An important point to apply the pre-QP-manifold to DFT is the fact that the condition 6) {{{{Θ, Θ}, f }, g}, h} = 0 (2.47) 5) If the bracket [·, ·] is graded antisymmetric, (2.46) is the graded Jacobi identity. In general, it is called the graded Leibniz identity. In this paper we simply call it the Leibniz identity 6) This condition is also derived in [20] as the condition on the pre-QP-manifold, and later in the context of an L ∞ algebra in [21].

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 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: 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, (2.58) In order to formulate the generalized Lie derivative of DFT by using the pre-QP-manifold, 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 , 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

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 We also define the corresponding DFT basis coordinates Q ′Â and P ′Â , 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 in each frame. We can, in principle, also consider the degree 2 functions including Q ′ and Q ′ . 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 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-QPmanifold as (3.14) 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: For the Hamiltonian function we obtain,

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 [22]. Especially, the n = 2 QP-manifold can be applied to the generalized geometry and the 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 ′ and P ′ , 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 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: 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 0 the standard   [8,26]. See also [28]. 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-QP-manifold.
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. Then, the canonical transformation e − π 2 δ U provides the GSS twist of the generalized vielbein EÂÎ(X) and the gauge parameter ΛÎ(X) of the reduced theory.
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 Thus, this derived bracket realizes the generalized Lie derivative of GDFT (2.31).
The closure condition for the derived bracket is provided by the weak master equation leads the following conditions for the generalized vectors and structure constants: 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: 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-QPmanifold is realized by the deformation of the Hamiltonian function.
DFT GSS generalized vector VM (X)P ′M VÎ(X) P ′  Here, we formulate the double geometry on a non-trivial background with pre-QP-manifold.
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 [29].

GL(2D) covariant formulation of pre-QP-manifold
Here, RMNŜR andRMNÎĴ are curvature tensor defined by ΓMNP and WMÎĴ , respectively: 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 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.
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.
1) The second condition (4.58) is satisfied by taking, 2) The second condition (4.58) can be satisfied by where the matrices κ and λ are defined as κMN = AÎM AĴN and λÎĴ = AÎN AĴN , 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. (4.68) The coordinate Ξ ∇ M is invariant under the canonical transformation.
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 , 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 j ′ * ( P ′ I ) = TÎ. Then, the derived bracket of P ′ I should reproduce the Lie bracket: The left hand side is calculated as,

Summary and Conclusion
In this paper we formulated the algebraic structure of DFT on a pre-QP-manifold in an We have also shown that the GSS compactification fits to this formalism. 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 Pstructure and the vielbein postulate.
We have shown that the familiar geometric objects are obtained from the pre-QP-manifold 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 [30] 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 [30].
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 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. In order to see the correspondence to original DFT structures, we restricted the transformation function EÂM to an element of O(D, D) and discussed the canonical transformation e π 2 δ E . For general θ case, we obtain the twisted Hamiltonian function in following form.

D Double Field Theory on group manifolds
Let us briefly recall the DFT on group manifold defined in ref. [29].

D.1 Background vielbein, covariant derivative and fluctuation
We consider a 2D-dimensional group manifold G and introduce local coordinates XM = The action of DFT WZW is rewritten using the generalized flux [29]. 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].