Metric algebroid and Dirac generating operator in Double Field Theory

We give a formulation of Double Field Theory (DFT) based on a metric algebroid. We derive a covariant completion of the Bianchi identities, i.e. the pre-Bianchi identity in torsion and an improved generalized curvature, and the pre-Bianchi identity including the dilaton contribution. The derived bracket formulation by the Dirac generating operator is applied to the metric algebroid. We propose a generalized Lichnerowicz formula and show that it is equivalent to the pre-Bianchi identities. The dilaton in this setting is included as an ambiguity in the divergence. The projected generalized Lichnerowicz formula gives a new formulation of the DFT action. The closure of the generalized Lie derivative on the spin bundle yields the Bianchi identities as a consistency condition. A relation to the generalized supergravity equations (GSE) is discussed.


Introduction
Recently, algebroid structures are being explored with the aim to characterize the effective theories of string geometrically in frameworks such as the generalized geometry, double field theory and exceptional field theory. In generalized geometry [1,2], we consider a generalization of the tangent vector, i.e., a generalized vector in T * M ⊕ T M over a given manifold M , and an action of O(D, D) as a rotation of the generalized vector. For a review see [3,4]. The formulation of the supergravity in generalized geometry setting is based on the structure of a Courant algebroid [5,6].
Double field theory (DFT) has been developed with the aim to formulate a T-duality invariant, gauge invariant theory [7,8]. The similarity between generalized geometry and DFT is well known and has been used to develop the theory from the early stage. See [9,10] for a review and references therein. The main difference is that in DFT the base manifold becomes twice the dimension of original manifold M , while in generalized geometry only the dimension of the fiber space is doubled.
The DFT picture is natural from the string point of view, since we consider the string moving in the dual manifoldM after T-dual transformation. Thus, DFT is defined on a 2D dimensional manifold M =M × M and the generalized tangent vector is a section of the tangent bundle T M. However, the algebraic structure on T M is not the standard Lie algebra of tangent vectors but an algebroid, which reduces to a Courant algebroid when reducing the DFT to the supergravity frame.
In standard DFT, usually a differential constraint on the fields is imposed to obtain the Ddimensional theory, called the section condition, which is associated to the matching condition of the spectrum in string theory. However, the section condition depends on the explicit choice of the local coordinates. Moreover, it is expected that non-geometric flux will rather be obtained by section-condition-violating configurations and, it is desirable to have the formulation based on the symmetry and independent of the section condition [11,10,12]. Of course, the dimension of the spacetime of the DFT is doubled and thus, eventually, we need to reduce the theory back to its original dimensional spacetime.
Basic notions of DFT in differential-geometric terms have been proposed in [13] where the author considers the 2D dimensional manifold M as a flat, para-Kähler manifold with a Courant-like bracket defined on its tangent bundle, which is called metric algebroid. The geometric aspects in DFT were also investigated in [14,15,16]. See also [17].
There is also an approach to DFT using the generalization of a QP-manifold or differential graded manifold [18] (see also [19,20] and reference therein). In ref. [21], it has been pointed out that the bracket structure of DFT can be obtained by using the differential graded manifold method. Then, the master equation was relaxed, which is called a pre-QP manifold, and the consistency condition was derived as a weak master equation [22,23]. When the master equation is relaxed, we are dealing with a metric algebroid.
In the pre-QP-manifold approach to DFT it is natural to analyze the Bianchi identities of the fluxes from the point of view of the weak master equation [24]. The QP-manifold approach is a kind of BRST-BV approach and the master equation is related with the closure condition of the underlying algebroid. From this point of view, we can say that in our previous paper, we confirmed that the Bianchi identity of DFT can be obtained from the condition of closure on the metric algebroid. Furthermore, in this analysis we found a pre-Bianchi identity, which gives the consistency of the algebroid of DFT before imposing the weak master equation [24]. Besides being consistent with the standard DFT, this formulation can also include more structure on the base manifold, e.g. group manifolds, as discussed in [25,26].
Recently, in the generalized geometry framework some developments to include the dilaton have been worked out in [27,28,29]. One method is to use the divergence operation in a Courant algebroid [30] to characterize the dilaton in the framework of generalized geometry. The authors used the derived bracket formulation by the Dirac generating operator [30]. This formulation can be understood as a quantization of the graded Poisson structure of the QP manifold [31].
Our motivation in this paper is to apply the Dirac generating operator (DGO) formulation to DFT, which will provide us with a mechanism to include the dilaton into the theory. However, unlike in generalized geometry, we will not require the square of the DGO to be a function, which is the analog of the relaxation of the master equation in the pre-QP manifold approach. This strategy will lead us to a relation between DGO and the pre-Bianchi identities which we can use to characterize the class of metric algebroid underlying DFT. We give a generalized Lichnerowicz formula for DFT and show that it is equivalent to the condition that the pre-Bianchi identity is satisfied. From the projected Lichnerowicz formula we derive an action for DFT. From the closure condition of the generalized Lie derivative on the spin bundle we obtain the Bianchi identities including the dilaton contribution.
The organization of this paper is the following: In section 2, we give a brief overview on Courant algebroid, metric algebroid and the Jacobi identities involved.
In section 3, we formulate the metric algebroid underlying DFT and the base independent form of the relations of bracket and anchor is discussed. Then, the generalized curvature tensor in this metric algebroid is constructed which enjoys tensorial properties. We derive the pre-Bianchi identity in curvature and torsion, and the pre-Bianchi identity for the dilaton. Formulae for rotation invariance of the frame are given. A generalization of the anchor map is also discussed.
In section 4, the derived bracket formulation by Dirac generating operator and an explicit expression for this Dirac operator are given. Then, the requirement of the pre-Bianchi identity on the structure functions is reformulated as the statement that a generalized Lichnerowicz formula holds.
The curvature scalar which appears in this generalized Lichnerowicz formula coincides with the scalar curvature obtained by taking the contraction of the generalized curvature tensor constructed in section 3.
In section 5, we introduce a Riemann structure by a splitting of the vector bundle. Then we define the Dirac operator compatible with the projection. The projected generalized Lichnerowicz formula is defined, and a proposal for a DFT action of the NS-NS sector is given.
In section 6, closure properties and Bianchi identities are analyzed.
In section 7, conclusions and discussions are given. A connection to the generalized supergravity equations (GSE) via the structure function F A is proposed. 2 Courant algebroid and metric algebroid structure Before we start discussing the metric algebroid, we briefly recall here the definition of a Courant algebroid [32] for convenience. We follow [33]. shows that the bracket is not necessarily anti-symmetric up to a derivative term. The last identity is the Jacobi identity in form of the Leibniz rule of the bracket. Since the bracket is not necessarily anti-symmetric, it is not equivalent to a cyclic form of the Jacobi identity, in general. Throughout this paper, Jacobi identity means the Jacobi identity of Leibniz form, unless we state differently.

Preliminaries: Courant algebroid
From the above defining equations, various properties of the bracket can be derived. Important formulae are where the bracket [−, −] L is the standard Lie bracket on T M .
The identity (d) shows that the bracket is a derivation w.r.t. the second argument, which follows from (a). (e) can be proven from (d) together with (b). The identities (f) and (g) are the consequence of (e) and (h), while (h) itself is the consequence of the Jacobi identity (c). The relations (f)-(h) will be discussed below.

Metric algebroid
In this paper, we use the following metric algebroid as the underlying symmetry structure of DFT.
We consider a vector bundle E → M . As in the Courant algebroid case, we introduce a bracket In order to discuss the correspondence with DFT, we need to introduce a set of local basis vectors where η AB is a symmetric constant tensor. We introduce η AB by η AB η BC = δ C A and the raising and lowering of indices by η. In this basis we can write the differential operator as (2.12) We then define a structure function F AB C ∈ C ∞ (M ) of the bracket by Using this basis, we can show that is totally antisymmetric.
Proof : For any a, b ∈ Γ(E), applying the second equation in the definition to [a + b, a + b], we obtain In the basis E A , E B = η AB = const, the r.h.s. is zero and therefore, F ABC = −F BAC . Furthermore, by the compatibility with the fiber metric we have and therefore F CAB = −F CBA . Combining the above two relations, we obtain F ABC = −F BAC = F BCA , i.e., F ABC is cyclic symmetric and thus totally anti-symmetric.

Jacobi identity
Since in the metric algebroid we do not require the Jacobi identity we define here a quantity which traces the deviation of the Jacobi identity from the Courant algebroid. For this purpose we define the following maps L : These quantities satisfy the following relations: (2.20) These relations follow from (2.15).
Note that these maps are not C ∞ (M )-linear in all arguments. Explicitly, one obtains and = a, b ρ(∂f ) .

(2.22)
We can rewrite the above expressions in a more symmetric form by considering the following map: (2.23) Then, the tensorial property is given by

Jacobi identity on TM
From the definition of L and L ′ and using the Jacobi identity of the Lie bracket [−, −] L we obtain the following relation: (2.29) Proof : The relation (2.28) can be shown by taking the anchor of L(a, b, c):  This means that the r.h.s. of (2.28) also vanishes and is another relation between the structure functions.

E-connection
On the metric algebroid E we define an E-connection, ∇ E : Γ(E) × Γ(E) → Γ(E) compatible with the inner product −, − . The E-connection is defined by the standard relations: for a, b, c ∈ Γ(E) We also require compatibility with the inner product: Using the basis E A , the connection ∇ E is defined by where W ABC is a gauge field. In the following, we also use the abbreviation ∇ E A = ∇ E E A as long as it does not cause confusion. Compatibility with the fiber metric yields that W ABC is antisymmetric in the last two indices:

E-torsion
Having defined the E-connection, one can introduce a corresponding E-torsion by The three terms in the first bracket correspond to the definition of the standard torsion except that the bracket is now the (Dorfman type) bracket of the metric algebroid. It is not C ∞ (M )-linear w.r.t.
the first argument a, a property which is recovered by the last term [30,35]. The same torsion was also introduced in DFT context in [36].
Using the local basis, we obtain the E-torsion in the form including the structure function as: Proof : From the definition of the torsion we see that given an E-connection ∇ E , we can define a new connection ∇ ′E as which defines a torsionless connection, since the torsion T ′ of this new connection vanishes as In other words, the connection W ′ ABC defined in (2.40) is independent of the torsion of the original E-connection and defines an equivalence class of connections up to a totally anti-symmetric part [27].
The bracket of the standard DFT is the D-bracket defined for X, Y ∈ T M as where indices are raised and lowered by η M N and η M N . It is easy to see that the D-bracket satisfies the axioms of a metric algebroid: and [X, X] D = 1 2 (∂ N X, X )∂ N .

DFT condition
In the present formulation, we consider a metric algebroid (E, [−, −], ρ) of a vector bundle E over M with local basis E A ∈ Γ(E). The bracket is characterized by the structure function F AB C . The The specific property of the present metric algebroid compared to a general metric algebroid is that the anchor is invertible.
Note that in this paper we consider no internal symmetry, therefore dim(E) = dim(T M).
On the tangent bundle we denote the inner product by −, − T M , and require for a vector field a ∈ Γ(E) to satisfy is the metric on the base manifold which is not required to be constant. We also assume that η AB is an O(D, D) metric so that the metric algebroid consistently includes the generalized geometry.
To summarize, we are considering a specific metric algebroid satisfying the following conditions: 1. The metric algebroid E on the manifold M with dim(E) = dim(T M).

The anchor ρ(E
3. There exists an inner product on T M, s.t. ρ(a), ρ(b) T M = a, b . We also assume that for the local basis, We call the above set of conditions the DFT condition. The standard DFT satisfies the DFT condition but additionally requires the vielbein E A M to be an O(D, D) element.

Jacobi identity with DFT condition
In the previous section, we introduced the maps L and L ′ in a general metric algebroid which trace the deviation from the Courant algebroid. Here, we discuss the Jacobi identities with DFT condition.
To make expressions more compact we consider the map φ :  We also introduce the structure functions corresponding to these maps by using the action on the local frame E A : where they are represented by the above maps as It is easy to see that the structure function φ ABCD is totally antisymmetric and represented by the structure function F ABC as: From (2.20) it is clear that the structure function φ ′ ABC is antisymmetric in the first two indices: On the metric algebroid the function φ(a, b, c, d) is not C ∞ (M)-linear w.r.t. all arguments, as we have seen in (2.24), nor is φ ′ (a, b, c). From (2.22) its transformation rule is obtained as: On the other hand, the following mapφ : Γ(E) ×4 → C ∞ (M ) is a tensor and totally antisymmetric: (3.14) We can show that the transformation of the extra terms cancel the ∆φ in (2.24) and recover the tensorial property ofφ. Now, we can prove the antisymmetry property of the mapφ(a, b, c, d) by evaluation in the local basis:φ Since φ ABCD is totally antisymmetric and φ ′ ABC = −φ ′ BAC , it is clear that the r.h.s. is totally antisymmetric and thusφ(a, b, c, d) is totally antisymmetric in all arguments.
While in a Courant algebroid, the condition φ ABCD = 0 yields the Jacobi identity on the structure functions F ABC , in the metric algebroid this condition is not covariant and depends on the local frame. However, as we have seen above, in the metric algebroid the quantityφ ABCD is a covariant tensor. Here, we use this property to characterize our metric algebroid bỹ This defines a class of metric algebroid which also includes the standard DFT. 5) It is also remarkable that in our previous analysis of DFT using the pre-QP-manifold, the local form of the condition (3.16), i.e.φ(E A , E B , E C , E D ) = 0 was obtained as a pre-Bianchi identity [24], and we refer to the above condition as the pre-Bianchi identity, the justification of which will follow. In the local basis, the pre-Bianchi identity is expressed as Since the Jacobi identity holds, J(a, b, c) = 0, by using the definition of the structure functions we Taking the inner product with ρ(E D ) we obtain for φ ABCD The same identity can be obtained by using the local basis and the structure functions F ABC and φ ′ ABC . We define the structure function of the Lie bracket, a generalized geometric flux F ′ ABC , as CD]E = 0 due to the section condition, and the pre-Bianchi identity reduces to the Bianchi identity φABCD = 0 (3.16).
Then, in the local basis we obtain the condition on this structure function F ′ ABC from J in (2.29) as where means the sum over cyclic permutation of indices. Thus, the tensor J ABCD can be expressed by the structure function F ′ ABC as One can easily show that this is equivalent to (3.20) by substituting the definition of F ′ ABC in (3.21).
Thus, the Jacobi identity J ABCD = 0 gives a condition on the structure functions.

Generalized curvature on metric algebroid
With DFT condition, we can still apply the same definition of the E-connection and E-torsion as in a general metric algebroid. On the other hand, the curvature has to be reconsidered.

Generalized curvature in DFT
We can define a curvature on a metric algebroid by completing the C ∞ (M )-linearity of the standard definition of curvature, similarly to the E-torsion. We start from the generalized curvature introduced in DFT in [36]: For convenience, we introduce the quantity R ∇ (a, b, c, d): The tensorial property of R HZ is where in the last line L ′ is the map defined in (2.18). The above equation means that if the Jacobi identity holds, i.e., in Courant algebroid L ′ h = 0 and in this case R HZ is a tensor.
On the other hand, in the metric algebroid we do not neglect the map L ′ , then the curvature R HZ does not have a tensorial property as shown above. However, with DFT condition we can Comparing the tensorial property of R HZ given in (3.28) and the transformation rule , it is easy to see that the following In the following, we refer to R : Γ(E) ×4 → C ∞ (M) given above as the generalized curvature in the metric algebroid. In the local basis E A , the explicit form of the generalized curvature tensor R ABCD is

Pre-Bianchi identity and curvature
Now, we are ready to discuss the covariant form of the Bianchi identity. In standard DFT there is a Bianchi identity given in terms of curvature R HZ and torsion [37,36]. Furthermore, in our discussion using the supermanifold approach [24] we encountered a corresponding pre-Bianchi identity.
Motivated by this, in the following we discuss the covariance of these identities as a structure in the metric algebroid.
We have defined a tensorφ and formulated the pre-Bianchi identity (3.16) which characterize the metric algebroid for DFT. We give here the pre-Bianchi identity in terms of the generalized curvature R defined above. This is achieved by realizing that the following identity holds: where ∇ E a is a connection on ⊗ 3 Γ(E) * by the Leibniz rule and A is an antisymmetrization map which defines a totally antisymmetric tensor for a map A(a 1 , a 2 , · · · , a n ) as AA(a 1 , a 2 , a 3 , · · · a n ) = where σ is a permutation.
The proof of (3.32) can be given by using a local basis E A , in which we have defined φ ABCD in (3.10). Using the explicit form of (3.31) (in local basis) and (3.27), we can show an identity which is similar to the one given in [36]: where we have replaced the E-torsion on the l.h.s. by the connection and the structure function using (2.38).
As we have seen in (3.28), R HZ on the r.h.s is not a tensor in the metric algebroid. Therefore, we use (3.29) to replace R HZ with the generalized curvature R including a correction term. Then, we see that by total antisymmetrization, this correction term combined with the other terms on the r.h.s. of (3.34) exactly produces the tensorφ ABCD . Thus, we get the identity (3.32) in the local basis. Since in the resulting expression each term is a tensor, we get the general form of the identity (3.32) which is independent of the choice of the frame. Now, imposing the pre-Bianchi identityφ(a, b, c, d) = 0 we obtain a frame independent formula which is a pre-Bianchi identity in curvature and torsion. As we saw in [24], the pre-Bianchi identity is the equation which holds when the flux is given by the vielbein as in standard DFT, that is, the standard DFT parametrization by the generalized vielbein is a solution of this pre-Bianchi identity.
Note that the identity of the form also holds. It means that this particular combination of the maps R HZ and φ is an element of -linear in all arguments, although each term on the l.h.s separately does not have this property.

Pre-Bianchi identity and dilaton
In DFT, there is another type of Bianchi identity which includes the contribution of the dilaton. We will discuss the property of the dilaton field using the divergence operator in the next section. Here, we focus on how the dilaton can be accommodated into this algebraic structure.
We start with the tensor J ABCD in (3.24). Taking a trace w.r.t. the last two indices we obtain the following tensor: To see the relation to the Bianchi identity including the dilaton discussed in standard DFT, we add a vector U A which satisfies As we see there is a non-trivial solution for this condition.
Now we can add this combination of U A to (3.37) and define a flux with one index by Using this flux F A we can rewrite the above identity as This identity is equivalent to the one including the dilaton in standard DFT [10,12]. In order to see this, we show that U A = 2ρ(E A )d satisfies above condition (3.38). Since the flux given as satisfies the pre-Bianchi identity (3.40). We postpone the discussion of the identification of U A as an ambiguity in the divergence and the relation between the pre-Bianchi identity for F A and the algebraic structure to the next section.
For reduction to standard DFT where the field d is identified with the dilaton and φ ′ reduces to the Weizenböck connection, we can show that the r.h.s. of (3.40) can be written as which coincides with the formula (1.5) in [12].

Rotation invariance of the frame
In this section, we check explicitly the local Lorentz covariance of the above structure functions, i.e., the covariance under the frame rotation, although it is rather apparent due to their tensorial structure. Thus, we consider the rotational group with dimension dim(E), and we denote an infinitesimal transformation using the basis E A as Then, the invariance of the inner product imposes Λ AB + Λ BA = 0, and the rotation is O(D, D).
The tensorial property of the structure function F ABC can be derived from the following relations: Therefore, evaluating in the basis, we get From this relation of the C ∞ (M)-linearity, we obtain the transformation of the structure function F ABC by identifying the functions f and g with the transformation parameter Λ AB as where Λ⊲ means the linear term of the transformation, i.e., Λ ⊲ F ABC is Similarly, we can get the transformation rules of the other quantities. In the following we list the transformations of the structure functions φ ABCD , φ ′ ABC and the maps L( for convenience: For the connection W ABC in (2.35) as is required for the spin connection under local Lorentz transformation. Note that the φ ′ BCA has the same transformation property as the connection W ABC .

Generalized anchor map
In standard DFT, the D-bracket is defined as (3.1), i.e., directly on the generalized vector X M ∂ M ∈ Γ(T M). This means that T M is identified with the metric algebroid. In the present formulation, we work with a metric algebroid on the vector bundle E. From this point of view, the metric algebroid T M is realized by the identification of the basis E A of the vector bundle E with a generalized vector of T M. For this it is convenient to introduce a generalized anchor map, which is a metric algebroid meaning that the basis is mapped as the anchor becomes trivial, i.e., 56) and the inner product is The bracket on the metric algebroid E is mapped to T M as By using the relation for the bracket [−, −] ϕ , we can evaluate the above relation in the basis: Then, we obtain that the original structure function F ABC can be expressed as where the function F ABC is given by We also introduce an affine connection ∇ T M by by employing the vielbein postulate as Reduction to standard DFT: The simplest case for this map ϕ is that the structure function and F ABC is totally antisymmetric, which is the generalized flux in standard DFT. Then the bracket

Derived bracket
In a Courant algebroid, the bracket can be represented by a derived bracket [38]. Here, we want to apply this formulation to the metric algebroid of DFT. Motivated by [30,27,39], we define the bracket as a derived bracket on a Clifford bundle Cl(E). Using the fiber metric, we can define a Clifford algebra by introducing a product Γ( anti-commutation relation among the elements a, b ∈ Γ(E) ⊂ Γ(Cl(E)) as 6) {a, b} = ab + ba = 2 a, b .
(4.1) 6) In principle, we have to distinguish an element a ∈ Γ(E) and its Clifford action on a Clifford module γ(a) where γ : Γ(E) → Γ(Cl(E)). To simplify the notation, we identify Γ(E) and γ(Γ(E)) and do not write this action explicitly.
We consider a connection ∇ Cl : Γ(E) × Γ(Cl(E)) → Γ(Cl(E)) on the Clifford bundle induced by a and imposing Leibniz rule w.r.t. the Clifford product. Then, the compatibility with the fiber metric holds: The Clifford algebra can be considered as a quantization of a graded symplectic manifold or, equivalently, a QP-manifold [31]. In [24], we have shown that certain algebraic relations in DFT can be for- A spin bundle S is a module over the Clifford bundle. We introduce a connection on S, for e ∈ Γ(E), χ ∈ Γ(S) and f ∈ C ∞ (M ). It satisfies the standard property of a connection as  It follows that for an element a ∈ Γ(Cl(E)) ∇ S e aχ = (∇ Cl e a)χ + a∇ S e χ . (4.10) Since the degree of ∇ S a is even and the graded bracket in (4.8) is an ordinary commutator, (4.10) is a consequence of the Leibniz rule of the commutator. Now, we are ready to introduce an odd differential operator, i.e., the Dirac operator / / D. The bracket (4.12) generated by / D is called a derived bracket. From these relations, / D is an odd graded linear differential operator, which is called Dirac generating operator [30]. The concrete form of the Dirac generating operator / D is discussed in §4.2. The axioms of the metric algebroid can be derived using the above definitions and the Jacobi identity for the graded commutator:  If the last term is zero, the above relation gives the graded Jacobi identity of the derived bracket.

Derived bracket by Dirac generating operator
Compared to the formulation of DFT using the graded manifold approach [24], this part corresponds to the weak master equation.
Since we do not require the Jacobi identity in the metric algebroid, the last term gives a measure for the breaking of the Jacobi identity. We obtain the following representation of L and L ′ defined in (2.17) and (2.18), respectively, by the derived bracket:  The generalized Lie derivative on χ apparently satisfies the above Leibniz rule w.r.t. the Clifford action: Note that there is an ambiguity to add a function. In particular, the closure of the Lie derivative on spinor yields

Dirac generating operator
In this section, we give a concrete form of the Dirac generating operator using a local basis. Namely, we construct a Dirac operator which satisfies the conditions (4.11),(4.12) and (4.13).
We use the standard representation of the Clifford action defined by We also use a zero connection ∂ A : Γ(S) → Γ(S) defined by {∂ A , γ B } = 0 , (4.28) where |0 ∈ S is a pure spinor, see appendix (A.3), and f ∈ C ∞ . Since the metric η BC is constant, (4.28) is compatible with (4.26).
A general form of the Dirac generating operator is given by the following odd differential operator where Then, it is straightforwards to show that / D satisfies conditions (4.11), (4.12) and (4.13). The structure function F A is an ambiguity of the Dirac generating operator, i.e., the metric algebroid is independent of the choice of the structure function F A . 7) 7) Note that if we have another Dirac generating operator/ D which satisfies the conditions (4.11),(4.12) and (4.13), A representation of the connection ∇ S on χ ∈ Γ(S) is specified by the action on the pure spinor |0 . The relations (4.8) and (4.9), are realized by taking 33) or equivalently, by defining the connection as The Dirac operator with this connection is Therefore, using the connection ∇ S we can write the Dirac generating operator as From the metric algebroid point view, the last term which is proportional to γ A is an ambiguity.
We use it in such a way that F A coincides with the trace of the connection, i.e., F A = W B BA and A A = 0, which is convenient from the point of view of DFT.
Using the definition of the E-torsion, / D can be written as This form shows that the Dirac generating operator is the Dirac operator with torsion free connection W ′ ABC in (2.40). As in [27], the Dirac generating operator is characterized by the structure functions F ABC and F A , thus the E-connection in (4.37) is not determined uniquely.
In the standard DFT, the Dirac operator in the same form as (4.30) is used in [12] where the structure functions F ABC and F A are replaced by F ABC and F A to formulate the action of the Ramond-Ramond sector [40,41,42]. It is also used to formulate the Ramond-Ramond sector of DFT on Drinfeld double [43].  Thus, since / D is a odd graded operator, we are free to choose FA ∈ C ∞ , i.e., / D −/ D ∈ Γ(E).

Generalized Lichnerowicz formula and pre-Bianchi identity
From the metric algebroid point view, DFT belongs to a class which is specified by the pre-Bianchi identity. We show that the conditions corresponding to the pre-Bianchi identity can be derived by using a generalized Lichnerowicz formula. The Lichnerowicz formula is a relation formulated by the difference of the square of the Dirac operator and a Laplace operator, cancelling the differential operators. Here, we define the generalized Lichnerowicz formula for the metric algebroid using the Dirac generating operator given in (4.37) which is induced by the E-connection but torsion free.

Divergence on spin bundle and Laplace operator
In order to define the Laplace operator, first we introduce here the divergence operator on the spin bundle. We define a contraction ι where e 1 , e 2 ∈ Γ(E), χ ∈ Γ(S).
For a given E-connection, we can define a divergence div ∇ by using a local basis as It is clear that this satisfies the above condition (4.41). In the appendix we show that the following div U ∇ also satisfies the above condition of a divergence (4.41): where U ∈ Γ(E), showing the degree of freedom in the divergence. The Laplacian of the given Since the divergence has an ambiguity, the Laplacian has also an ambiguity of U ∈ Γ(E).

Generalized Lichnerowicz formula
The Laplace operator which appears in the generalized Lichnerowicz formula is the one associated to the connection ∇ φ ′ defined by φ ′ (a, b, c) in (3.6). We have shown that the structure function φ ′ ABC has the same transformation property as the connection W CAB , thus we introduce a connection on the Clifford module Γ(S) s.t.
The corresponding Laplace operator is then given by where U A is a vector representing the ambiguity in the divergence as discussed previously. The definition of the generalized Lichnerowicz formula is the square of the Dirac generating operator with derivative terms covariantly subtracted: where J BCD D is the tensor defined in the identity given in (3.37). The term proportional to γ ABCD is the tensorφ ABCD in (3.15) which gives the pre-Bianchi identity.
The derivative terms of the square of the Dirac operator appear in both, the scalar part and the part proportional to γ AB . The Laplacian ∆ φ ′ is chosen such that the terms containing the derivative operator in the part γ AB cancel. However, the derivative term in the scalar part remains as shown Finally the scalar part is given by the generalized Riemann scalar R constructed from the gen- . Therefore, R ∇ in the generalized Lichnerowicz formula which is written with the connection W ′ , is represented by the structure functions as where the ambiguity of the Dirac generating operator is identified as F A = W ′B BA = W B BA As we discussed, DFT is realized on a metric algebroid where the pre-Bianchi identities vanish.
For this class of metric algebroid, we have the following generalized Lichnerowicz formula: The above result can be put into the following statement: The requirement that the pre-Bianchi identities for the structure functions hold can be rephrased as the requirement that the generalized Lichnerowicz formula is satisfied. Note that the generalized scalar curvature does not vanish in general.

Action from Dirac generating operator
To construct an action using the above approach, we propose a projected generalized Lichnerowicz formula which is consistent with Riemannian structure on the metric algebroid.

Riemannian structure
The splitting of the vector bundle in DFT and in generalized geometry has been worked out in great detail, as can be found in [44,45,46,37,36,5].
It is known that the metric structure on DFT can be introduced by splitting the vector bundle E into positive and negative sub-bundle V ± as in the generalized geometry: where V + and V − are orthogonal to each other.
Using the projection operators P ± : E → V ± , any vector can be split into V ± as a = a + + a − where a ± = P ± (a). The sub-bundles V + and V − are orthogonal and thus the inner product can be split as The generalized metric is a positive definite product defined for a, b ∈ E by

Compatible connection
As in the generalized geometry, we consider the E-connection patible with the splitting. Using the local basis E A ∈ Γ(E), compatibility requires From this we conclude that the nonzero components of the connection are W Aab and W Aāb .
In the following, we construct an action from the Dirac generating operator. As we discussed, the Dirac generating operator is free from torsion as given in (4.37) and thus we can choose the torsionless connection without loosing generality 8) . Then we get the relation between the structure function and connection as i.e., the totally antisymmetric part of the connection is defined by the structure function.
From the definition of the torsion (2.37), for the mixed argument we get (5.12) and similarly for T (a + , b − , c − ). Thus the mixed part of the torsionless compatible connection is given by 13) which is known as the generalized Bismut connection in generalized geometry [2,3]. From this, we conclude that the mixed part of the compatible connection is completely defined by the structure function as Wā bc = Fā bc , W abc = F abc . On the other hand, the pure part of the spin connection is not completely defined by the structure functions in the Dirac generating operator except for the totally antisymmetric components: (5.15) and the trace part

Projected Dirac operator and Laplacian
In the following we formulate the action using the generalized Lichnerowicz formula with the above compatible connection. For this we introduce here the projected Dirac operator and the Laplacian.
(See appendix for details.) Then, we consider the following projected connections Since ∇ S + a contains the only mixed type E-connection, it can also be written by the structure function as As in the generalized Lichnerowicz formula, we further have to consider the connection induced by φ ′ ABC on the Clifford module Γ(S + ) s.t.
The corresponding Laplace operator is given as

Projected Lichnerowicz formula and DFT action
The action of DFT can be formulated by the following projected Lichnerowicz formula as The first two terms are the analogous combination appearing in [5] where the supergravity is formulated using the generalized geometry. The difference is that the first two terms here contain the differential operators.
By using the projected Dirac operator / D + in (5.20), the first term is The second term is the divergence of the projected connection ∇ S + − given in (5.19): 26) where div ∇ of the projected connection is given in the appendix. The last term is the Laplacian from the projected connection ∇ φ ′+ given in (5.23): As in the generalized Lichnerowicz formula derived in §4.3.2, the last term cancels the differential operators in the first two terms keeping the covariance. For this we identify the vector field U A ,i.e., the ambiguity in the divergence, as in the case of the generalized Lichnerowicz formula as The result is where By using the identification of the structure functions F abc , F A and spin connection W Abc , we can write the above R DF T as where the R ABCD is the generalized curvature in the metric algebroid defined in (3.30).

The above action is O(D) × O(D) covariant. Up to the section condition, we can see that the
R DF T is proportional to the standard DFT action if we substitute F ABC = F ABC and φ ′ ABC = Ω CAB , which is a solution of the pre-Bianchi identity (3.17). We also identify the dilaton by (3.42), i.e., the which is also a solution of the pre-Bianchi identity (3.40). Now, we can formulate the action of the DFT using the above projected Lichnerowicz formula.
where C = 0 ′ |0 ′ is a constant and µ 0 is defined in appendix C.
Applying this formulation to the standard DFT, S inv becomes equivalent to the standard action as follows. First, we construct a concrete representation of Γ(Λ L a (f µ as follows: where L a is the standard Lie derivative on Γ(Λ where µ 0 is identified with the integration over M. We can prove that S inv is invariant under the generalized Lie derivative as follows, The transformationδ a can be identified with the gauge transformation of the dilaton in the standard DFT and in this way e −d is considered as a half density. The gauge transformation of the field can be discussed by choosing a concrete form of the structure functions F ABC and F A , and discussing the failure of the covariance. For example, by taking the standard DFT solution of the pre-Bianchi identity, i.e. F ABC = F ABC and F A = F A , it is known that the action is gauge invariant (see for example [10]).

Closure and derived bracket
In the formulation of a Courant algebroid using the Dirac generating operator, the closure of the bracket, i.e. the Jacobi identity, is realized by requiring that the square of the Dirac generating operator is a function [30]. Here, we are considering a metric algebroid, i.e., the Jacobi identity is not required for the derived bracket and thus the square of the Dirac operator is not necessarily a function. On the other hand, since the gauge symmetry of DFT is generated by the generalized Lie derivative [7], the closure of the D-bracket on the fields, which is the gauge consistency constraint discussed in [12], is important.
From the point view of a metric algebroid, the gauge consistency constraint can be discussed after solving the pre-Bianchi identity, i.e., we have to represent the fluxes in terms of the fundamental fields such as generalized dilaton and generalized vielbein. This opens up a number of possibilities, as we will indicate below, however, the detailed study of them is beyond the scope of this paper.
Therefore, in this section we show how our formulation produces Bianchi identities and consistency constraint corresponding to the standard DFT case.

Closure on E
For the closure of the generalized Lie derivative in the present formulation, we have to require L(a, b, c) in (4.20) to vanish. Note that vanishing of (4.21) on the Clifford bundle follows. Thus, we require the following closure condition which corresponds to the weak master equation in the supermanifold approach We get the explicit form of closure constraint as follows This condition is understood as a constraint for the structure function F ABC and section Γ(E). In principle, we can seek for the solution where F ABC and φ ′ ABC satisfy a relation with the coefficients a A , b A , c A and their derivatives. However, for application to DFT in mind, we are interested in the case where the basis of Γ(E) satisfies the condition Then, we get the Bianchi identity for F ABC , i.e., where φ ABCD is given in (3.10). For the other terms to vanish we require for a set of c ∈ Γ(E) satisfying the following constraint on the coefficients From (6.7), which we call the closure constraint, we obtain a restriction on the space of sections Γ(E) and we denote this subset as Γ(E) ccE .
To summarize, the closure of the generalized Lie derivative requires the vanishing of the condition (6.1) which we call closure constraint. If we require that the square of the DGO is a function, of course, this condition is satisfied. However, here we consider that this condition restricts the structure functions and the space of sections in Γ(E), like the weak master equation in the supermanifold approach.
To apply the above formalism to DFT, we require that the basis of Γ(E) satisfies the closure condition (6.1), then this condition implies the Bianchi identity for F ABC and defines Γ(E) ccE via the closure constraint. In standard DFT, the generalized anchor is applied on (6.4), then we obtain the constraint (6.7) where the coefficients of a, b, c are replaced by the components of the vielbein, which is equivalent to the constraint given in [12][10].
Note that, as we discussed, the Bianchi identity φ ABCD = 0 depends on the choice of the basis.
However, together with the conditions (6.7) and (6.8), the covariance w.r.t. the rotation of the local frame is recovered. Note also that we do not get the Bianchi identity for F A from closure condition (6.1), which we postpone to the next section.

Closure on S
We have defined the generalized Lie derivative on S (4.24) with which we can require the closure condition of the generalized Lie derivative on S as where a, b ∈ Γ(E), χ ∈ Γ(S). Similar to the closure condition on Γ(E), eq. (6.9) is too strong on arbitrary elements χ. Therefore, we interpret it as a restriction on Γ(S). We define a subspace Γ(S) cc ⊂ Γ(S) and Γ(E) cc ⊂ Γ(E) whose elements satisfy the above closure condition. Note that by Γ(E) cc , we can also define Γ(Cl(E)) cc . For consistency, Γ(S) cc must be a representation of Γ(Cl(E)) cc , i.e., ∀ a ∈ Γ(Cl(E)) cc , ∀ χ ∈ Γ(S) cc =⇒ aχ ∈ Γ(S) cc . This condition is equivalent to the one for Γ(E) ccE defined in the previous section.
The explicit form of the closure condition for χ is Here, we do not solve this condition in full generality. Instead, we give one example which connects to the standard DFT. Assume that γ A is a solution of the closure condition as in the discussion on the closure on Γ(E), i.e., γ A = E) A . In this way we can get the Bianchi identity for F A as follows.
The same discussion as in the previous section applies which gives us the Bianchi identity for F ABC and the closure condition on Γ(E), i.e., (6.5), (6.7) and (6.8). Using these equations, the closure constraint on Γ(S) reads This condition is understood as a constraint on the structure function F A and the section Γ(S).
To obtain the Bianchi identity, we consider the special case where the base satisfies the closure constraint on Γ(S). Taking a = γ A , b = γ B ∈ Γ(E) cc , we obtain Then we require the closure also for general elements a, b. First, taking b = γ B in (6.14), we get the following relation Using this equation we finally obtain for general elements a, b Thus, the restriction of the vector bundle Γ(E) ccE ⊂ Γ(E) is not enough to satisfy the closure condition on Γ(S), i.e., Γ(E) ccE ⊂ Γ(E) cc . In standard DFT, this condition is satisfied by the strong Furthermore, assuming that |0 ∈ Γ(S) cc , we get the Bianchi identity for F A by the equation where f is a function in Γ(E) cc . In this case, Γ(S) cc becomes Γ(S) cc = {O |0 |O ∈ Γ(Cl(E)) cc } . (6.20) To summarize, in the case where γ A ∈ Γ(E) cc and |0 ∈ Γ(S) cc , the closure on Γ(S) requires In this way, we obtain the Bianchi identities for F A and F ABC . In standard DFT, the Bianchi identity is solved by imposing the strong constraint. In this case η M N is constant and a solution of the Bianchi identity is given by F ABC = F ABC in (3.61) and F A = F A in (5.32).

Conclusion and discussion
In this paper, after giving a brief survey on the algebraic structure of a metric algebroid, we analyzed the properties of the structure functions relating to DFT. By requiring independence of the choice of the local bases, we found that a pre-Bianchi identity can be obtained as a completion of the Bianchi identities. As a result we obtain the mapφ the vanishing of which is a pre-Bianchi identity.
The In the second part of this paper, we gave a formulation of DFT using the Dirac generating operator (DGO). Unlike in generalized geometry, we did not require the square of the DGO to be a function. This relaxation of the condition on the DGO lead us to the structure of a metric algebroid.
In this setting the DGO is the fundamental object and its square contains differential operators in general. This investigation gave several new insights.
From the square of this DGO with derivative terms covariantly subtracted we derived the pre-Bianchi identities with which we have characterized the metric algebroid underlying DFT. After the subtraction, the square of the DGO contains three contributions, and requiring the result to be a scalar function we obtain both pre-Bianchi identities, i.e. (3.17) and (3.40), and the scalar part becomes the scalar of the covariant generalized curvature (4.48). This procedure results in a generalized Lichnerowicz formula. Thus, the condition for the pre-Bianchi identities to hold is equivalent to the condition that the generalized Lichnerowicz formula is satisfied. Given a metric algebroid, there is an ambiguity in the DGO, and this freedom allows to introduce the dilaton into the structure function F A .
To obtain the action, we introduce a Riemann structure by a splitting of the vector bundle into positive and negative subbundle. By using the corresponding projection we obtain the projected generalized Lichnerowicz formula, which is proportional to the projected generalized scalar curvature under the pre-Bianchi identity. Then, we propose an action for DFT in terms of the projected Lichnerowicz formula. To formulate the measure of the action, we introduced the inner product of the pure spinor µ = 0|0 which is O(D, D) invariant. We could interpret µ as the measure dX 1 ∧ · · · ∧ dX 2D e −2d in the standard DFT choosing the representation. However, µ may not be in ∧T M in general.

Remarks on generalized supergravity equations (GSE)
Recently, a generalization of supergravity, originally proposed in [47,48], is discussed by several authors as a possibility to modify the supergravity equations to a more general set of field equations in the context of integrable deformations, keeping consistency with superstring [49,50]. These integrable deformations are considered to be closely related to non-Abelian T-duality transformations [51,52,53] and also to Poisson-Lie T-duality [54,55].
One way to obtain the generalized supergravity equations (GSE) in DFT which fits to the approach given here is to consider a modification of the field representation of the structure function F A . This modification is possible due to the ambiguity X in the divergence compatible with the splitting V ± , which is discussed in [28] in the context of generalized geometry.
While in standard DFT the structure function is represented by F A = F A , by the ambiguity in the divergence the structure function F A can include a generalized Killing vector X as follows: where H is a generalized metric (B.32) in the appendix. Since the combination of the pre-Bianchi identity (3.17) is covariant but not necessarily zero, we may extend it by a covariant term as where the r.h.s. is an additional term corresponding to a derivative: The structure function F A = F A + X A is a solution of this covariant equation. This means that X can be interpreted as a freedom in the pre-Bianchi identity.
Furthermore, if X satisfies ∂ [M X N ] = 0, then the pre-Bianchi identity for F A becomes zero, which we required in this paper to characterize the metric algebroid for DFT. The simplest solution of this condition is X M = constant. In this case, the ambiguity X of the structure function F A becomes a constant Killing vector, which is used as an ansatz for the dilaton to obtain the GSE from DFT with non-standard section [56].

A.2 Spin bundle of V +
As in the previous section, we consider a spin bundle Γ(S + ) as a module of Γ(Cl(V + )). The vector bundle E can be written by the positive and negative subbundles E = V + ⊗ V − . Then, the Clifford bundles Cl(V ± ) can be defined by a bracket {−, −} for a, b ∈ V ± ⊂ Cl(V ± ), Now, assume that V + admits a Clifford module S + , i.e., V + can be split into D/2-dimensional isotropic subbundles L 1 and L 2 where V + = L 1 ⊕ L 2 . We define a basis l a , l a of L 1 , L 2 , respectively.
For a given E-connection, the corresponding divergence div ∇ is given by where div ∇ satisfies the relation (B.1), i.e., Note that due to the property of the covariant derivative, the summation over the basis does not depend on the choice of the basis. Using the local basis, we obtain Considering an arbitrary divergence div which satisfies the relation (B.1), the difference between div ∇ and div is a C ∞ (M)-linear function div(f a) − div ∇ (f a) = f (div(a) − div ∇ (a)) .
Thus, an arbitrary divergence can be written by To construct a Laplace operator on Γ(L) we define a connection ∇ L : An explicit form of this connection is Using these operators a Laplace operator ∆ : Γ(L) → Γ(L) can be defined as These definitions coincide with the divergence (B.6) on E. We will see that the standard Laplacian on functions is also included. In the following we show concrete forms of the divergence and the . This divergence div U ∇ is equal to the divergence (B.6). For U = 0 the Laplace operator is the standard Laplacian on C ∞ (M).

B.2.2 Laplace operator on S
In the case where L = S, we obtain where a ∈ Γ(E) and χ ∈ Γ(S).
where γ a is the basis of Cl(V + ) and ∂ A is a zero connection on S + , i.e., for f ∈ C ∞ (M) and the pure spinor |0 on S + , Using this connection, we can define the connection ∇ E⊗S + , the divergence and the Laplace operator where a ∈ Γ(E) and χ + ∈ Γ(S + ).

B.3 Compatible divergence
In this paper, a metric algebroid is defined on a vector bundle Γ(E) = Γ(V + ) ⊕ Γ(V − ), where V + , V − are positive and negative definite subbundles, respectively. In order to discuss a compatible divergence with V ± as in [28], we prepare some structures as follows. First, we reformulate the projection E → V ± in (5.6), defining two tensors η and H on E ⊗ E, where η AB and H AB are defined by the inner product −, − and the generalized metric (5.4), respectively. The projection tensor P ± ∈ E ⊗ E is Using the above tensor, we can write the projection E → V ± as where a ∈ E. The generalized Lie derivative on Γ(E) ⊗ Γ(E) is defined by the Leibniz rule: Then, the generalized Lie derivative satisfies the Leibniz rule also for the contraction ι a where a, b, c, d ∈ Γ(E).
From this definition we can show To see this we calculate L a b where we used ι a η = a. Since b ∈ Γ(E) is an arbitrary element, this means that (B.37) holds.
where we use L a η = 0. Since b ∈ Γ(E) is an arbitrary element in Γ(E), we get This condition means that X is a generalized Killing vector. In the case where we use this compatible divergence for the Laplace operator ∆ φ ′+ in the DFT action, the structure function F A becomes The similar condition has been considered in the generalized geometry in [28]. The generalization considered here has an ambiguity. In the generalized geometry, a special divergence div µ is used instead of div 2∂d ∇ φ ′ . div µ is defined by a D-form in ∧T * M , div µ a = µ −1 L ρ(a) µ , (B.47) where L is the standard Lie derivative. div µ satisfies div µ [a, b] − ρ(a)div µ b + ρ(b)div µ a = 0 , (B.48) where a, b are elements of a Courant algebroid and [−, −] is the bracket of this Courant algebroid.
On the other hand, in DFT, div 2∂d ∇ φ ′ does not satisfy such a condition. However, div 2∂d ∇ φ ′ can be identified with div µ in the supergravity frame as follows: We use the ansatz of the vielbein E a = −e m a B mn dX n + e n a ∂ n , E a = e a n dX n , where e n a , B mn are identified with the vielbein and the Kalb-Ramond field, respectively, in generalized geometry. We obtain div µ E a = ∂ n e n a − 2e n a ∂ n d , div µ E a = 0 , (B.49) where µ = e −2d dX 1 ∧ · · · ∧ dX D . On the other hand, div 2∂d where E a = −e m a B mn ∂ n + e n a ∂ n , E a = e a n ∂ n corresponding to E A ∈ T M ⊕ T * M and φ ′ AB C = Ω C AB .
Thus, div 2∂d ∇ φ ′ can be identified with div µ in the supergravity frame in standard DFT. But the identification of µ has an ambiguity corresponding to the dilaton shift. In generalized geometry this ambiguity in the choice of µ is not important since it changes X by total derivative and thus the condition of the compatible divergence is the same. On the other hand, in DFT this ambiguity changes the condition of the compatible divergence, since the generalized Killing vector X shifted by the total derivative is not a generalized Killing vector, in general. So, the generalization of the compatible divergence is not unique. Here, this ambiguity is fixed by the condition (B.41).
Similarly, we can also identify 0 ′ |γ A 1 · · · γ An |0 ′ with a constant scalar, since the generalized Lie derivative on γ A 1 · · · γ An |0 ′ is given by 1 Furthermore, we can show that the line bundle Λ ′ can be identified with C ∞ (M) as follows. Since all elements on Γ(S ′ ) can be generated by γ A , the generalized Lie derivative on an arbitrary inner product can be written by Under the identification of 0 ′ |γ A 1 · · · γ An |0 ′ with constant scalar, this transformation is equal to the one for C ∞ (M). Thus, the spin bundle Γ(S ′ ) is characterized as the weight 0 by the existence of the inner product Γ(S ′ ) × Γ(S ′ ) → C ∞ (M).
Then, we identify the dilaton in µ in a base µ 0 ∈ Γ(Λ) as In order to recover the dilaton gauge transformation with a weight 1 2 , we introduce the gauge transformation of the dilatonδ a e −d generated by using the above transformation of µ Therefore, the gauge transformationδ of e −d is the one of a half density in the usual sense.