Extended Doubled Structures of Algebroids for Gauged Double Field Theory

We study an analogue of the Drinfel'd double for algebroids associated with the $O(D,D+n)$ gauged double field theory (DFT). We show that algebroids defined by the twisted C-bracket in the gauged DFT are built out of a direct sum of three (twisted) Lie algebroids. They exhibit a"tripled", which we call the extended double, rather than the"doubled"structure appeared in (ungauged) DFT. We find that the compatibilities of the extended doubled structure result not only in the strong constraint but also the additional condition in the gauged DFT. We establish a geometrical implementation of these structures in a $(2D+n)$-dimensional product manifold and examine the relations to the generalized geometry for heterotic string theories and non-Abelian gauge symmetries in DFT.


Introduction
Double field theory (DFT) is a field theory framework that encompasses T-duality in string theories [1][2][3].The theory is defined in the 2D-dimensional doubled space M 2D on which fundamental fields H M N (X), (M, N = 1, . . ., 2D) and d(X) are defined.Here H M N (X) and d(X) are the generalized metric and the generalized dilaton, respectively.The coordinate of the doubled space M 2D is decomposed as X M = (x µ , x µ ), (µ = 1, . . ., D) where xµ and x µ may be identified with the winding and the Kaluza-Klein coordinates.The theory exhibits manifest O(D, D) structure and possesses a gauge symmetry.In order to be physically consistent, all the quantities including fields and gauge parameters in DFT must satisfy a physical condition called the strong constraint.This indeed guarantees the gauge invariance of the action and closure of the gauge algebra.The strong constraint is trivially solved by quantities that depend only on x µ .In this case, the action of DFT reduces to that of the NSNS sector in type II supergravities.A remarkable fact about DFT is that the diffeomorphism and the gauge symmetry of the Bfield are unified in a T-duality covariant way.This is realized by the gauge symmetry in DFT whose algebra is governed by the C-bracket.
The algebraic structures defined by the C-bracket are known as algebroids.They are in fact a unification of geometries and symmetries.The C-bracket is quite different from the Lie bracket.Most notably, the C-bracket does not satisfy the Jacobi identity in general.This leads to the notion of the metric (DFT or Vaisman) algebroid [4,5].For the O(D, D) DFT, several algebroid structures have been studied [6][7][8][9][10][11][12].They are strongly tied with geometric structures of M 2D .It is discussed that the doubled space is realized by a para-Hermitian geometry in which the metric algebroid was first defined [13].The relations among the strong constraint, physical spacetime and the Courant algebroid in the generalized geometry have been studied [14][15][16][17].Among other things, doubled structures of algebroids play important roles in DFT.It is proven that the metric algebroid inherits a doubled structure [6].This is an analogue of the Drinfel'd double for Courant algebroids [18] (see for the Drinfel'd double for Lie algebras [19]).The Drinfel'd double is based on a pair of the dual (maximally isotropic) Lie algebroids (L, L) by which a Manin triple is defined.Indeed, upon imposing the strong constraint, the C-bracket becomes the Courant bracket and then the metric algebroid reduces to the Courant algebroid [6,20].The notion of the Drinfel'd double is necessary especially in the context of the Poisson-Lie (and non-Abelian) T-duality (and plurality) [21][22][23].The duality (plurality) is interpreted as the exchange of the Manin triples for given algebraic structures and it is a generalization of T-duality.In this sense, the Poisson-Lie T-duality (U-duality) is naturally understood in DFT and exceptional field theory (EFT) [12,[24][25][26][27][28][29][30][31].
On the other hand, for heterotic string theories and gauged supergravities, an alternative formalism of manifest T-duality known as the gauged DFT has been developed [32,33].In the gauged DFT, non-Abelian gauge symmetries are introduced by gauging a subalgebra of the extended duality group O(D, D +n).The gauge symmetry in DFT is modified and governed by the twisted C-bracket.The physical condition is then supplemented by an additional constraint which we will call the gauge condition.The advances of the gauged DFT are the appearance of non-trivial Yang-Mills sectors.Although T-and U-duality properties of gauged supergravities are studied in various contexts (see for example [34][35][36]), the doubled structure of the gauged theories are still unclear.
The purpose of this paper is to establish doubled structures of algebroids endowed with the twisted C-bracket in the gauged DFT.We will find that the algebroids in the gauged DFT admit Drinfel'd double like structures.We will show that the consistency of the doubled structure leads to the physical conditions including the strong constraint and the gauge condition in the gauged DFT.
The organization of this paper is as follows.In Section 2, we introduce the gauged DFT and its gauge symmetries.The gauge algebra is governed by the twisted C-bracket and the physical conditions in the gauged DFT are presented.In Section 3, we discuss a geometrical implementation of the twisted C-bracket.We will discuss a triple foliated structures in (2D+n)dimensional product manifold M 2D+n .We will then exhibit that a metric algebroid defined by the twisted C-bracket is constructed by three (twisted) Lie algebroids.We find that the consistency conditions for the Courant algebroids result in the physical conditions in the gauged DFT.In Section 4, we discuss the relations to the generalized geometry for heterotic string theories.Section 5 is devoted to conclusion and discussions.A mathematical glossary is found in Appendix.

Gauged double field theory and twisted C-bracket
In this section, we briefly introduce the gauged DFT and its gauge symmetries.We also introduce the notions of the non-Abelian gauge symmetry in DFT, the twisted C-bracket, the strong constraint and the additional gauge constraint.They will be key ingredients for algebroid structures discussed in Section 3.
We first start from the action of O(D, D + n) DFT; where H M N (X), (M, N = 1, . . ., 2D+n) and d(X) are the generalized metric and the generalized dilaton defined in the (2D + n)-dimensional doubled space M 2D+n for which the coordinate The standard parametrizations of the generalized metric and the dilaton are given by where g µν (X), g µν (X) are a symmetric D × D matrix and its inverse.An n × n symmetric constant matrix κ αβ and its inverse κ αβ and a D × n matrix A µα (X) and a scalar quantity ϕ(X) have been introduced.We have also defined the quantity and its inverse η M N .Note that the generalized metric H M N is an element of O(D, D + n).The action (2.1) is manifestly invariant under the O(D, D + n) transformation; Now we gauge a subgroup G of O(D, D + n) and break it down to O(D, D) × G.This is done by introducing a constant (2,1)-tensor flux F M N K such as Here F α βγ is the structure constant for the gauge group G whose dimension is dim G = n.The constant F M N K must satisfy the following relations; In order to keep the gauge invariance, the action (2.1) is deformed such as [32] where (2.9) The action (2.8) is invariant under the following gauge transformation; provided that the following physical conditions are satisfied; Here * are all the quantities in DFT including the generalized metric, the generalized dilaton and the gauge parameters Ξ M .The first condition above is just the level matching condition of closed strings and the second is known as the strong constraint in the context of the ordinary O(D, D) DFT.We call these the physical conditions.The last one is specific to the gauged version of DFT and we call it the gauge condition in the following.
The gauge transformation (2.10) is closed under the conditions (2.11), namely, for an arbitrary O(D, D + n) vector V M , we have where the left-hand side is the commutator of δ Ξ 1 and δ Ξ 2 .In the right-hand side, we have defined the twisted C-bracket; Note that the conditions (2.11) are trivially solved by quantities that depend only on x µ .In this case, the action (2.8) reduces to that of a gauged supergravity in D dimensions.Among other things, when D = 10 and n = 496 and G is SO (32) or E 8 × E 8 , the theory reduces to the heterotic supergravities in ten dimensions.

Extended doubled structure for algebroids
In this section, we introduce a geometric implementation of (twisted) Lie algebroids in the extended doubled space M 2D+n and discuss algebras associated with it.We show that the twisted C-bracket (2.13) in the gauged DFT is rewritten by the geometric quantities.We then discuss an extended doubled structure of the metric algebroid defined by the twisted C-bracket.We in particular focus on the compatibility conditions of algebroids and analyze the relation with the physical conditions in the gauged DFT.

Lie algebroids in M 2D+n
We assume that the (2D + n)-dimensional doubled space M 2D+n endowed with the pseudo-Riemannian metric η M N in (2.4) admits an integrable product structure.Namely, there exists an endomorphism P : T M 2D+n → T M 2D+n such that P 2 = 1 2D+n and its associated Nijenhuis tensor vanishes.The product structure P defines a rank n distribution L in T M 2D+n and its complementary pair D. They corresponds to P = +1 and P = −1 eigenbundles.Then we have the decomposition of the tangent bundle T M 2D+n = D ⊕ L. Since the distributions are integrable, there exists a natural coordinate system of M 2D+n ; X M = (X M , xα ) M = 1, . . ., 2D, α = 1, . . ., n such that D, L are spanned by the basis at each point in M 2D+n .Then the structure group O(D, D + n) is naturally introduced.In particular, the integral manifolds of D, L are defined by slices xα = const.and X M = const, respectively.We further assume that the 2D-dimensional manifold defined by xα = const.is a para-hermitian < l a t e x i t s h a 1 _ b a s e 6 4 = " b a k j r C P s c c k 7 S D W j z 8 h S h p w B X z d / N n 8 5 H 5 m / m L + a f 5 b y H W l D C 4 L a f 4 t I U s C 3 q X v n h 5 + 5 9 S q S E + I z h M p E o l B m g h 9 8 F Y M z / F y 5 G H W j 4 b Z 2 t h n y G v j o / 7 Z h j 7 X + / Z C G 1 8 j z z q I n d A F C 7 n i F k e f / 7 N o + 1 b W 1 e m r 5 s / m n + h l 3 8 w H 5 q / o p 9 H x 3 8 7 D 5 p s 6 e q L e r j n X q R r w S O 4 j j U s Z x X S 5 y M e S b x J l T r i / P 6 n W q h c v p W K H a o b 4 G x v B t / K 1 T p e t T b v c p g 0 7 x f 5 O 8 5 c p 6 o 7 7 6 u i E q 0 H K 5 z T N I l 9 c i t 3 T 6 h m e s 7 3 q J v u D M N Q r M s r p x l j p n s v b P r w d + c j y V + z T P g g F l 0 3 I r a b 7 P M 3 I n Q 2 l n 9 u + u B s 2 C + 8 Q l 5 O Z H + d z v p M a L U f L n k I f 3 / m P O N n / F 5 2 P / n 6 q U j 5 i t U P k o D r V 5 z D 2 q 1 q c F a I 0 c P j 3 q C D 6 2 2 9 c 6 N 5 6 4 5 4 8 3 where L, L are integrable distributions defined by the para-hermitian structure such that L = T F, L = T F.Here the base space F, which may be identified with the physical spacetime, is defined by xµ = const.and parameterized by x µ .The other F is defined by x µ = const.and parameterized by xµ .These structures allow us to find the foliation property for M 2D+n , namely, the coordinate of the base space is decomposed as X M = (x µ , x µ , xα ).A hypersurface M 2D defined by xα = const.contains leaves spanned by x µ , xµ .The n-dimensional gauge space is defined by a leaf x µ = const, xµ = const.for which it is endowed with a metric κ αβ .These structures define a tri-foliated space (Fig 1).Then a vector field Ξ on M 2D+n is decomposed as Given these properties, we define the Lie brackets in each subbundle by For later convenience, we define differential operators d, d, d for a function f in We now introduce totally anti-symmetric products of vector fields in L, L and L; where 1 ≤ p ≤ D for L, L and 1 ≤ p ≤ n for L. The operators d, d, d act on a p-vector field These are analogous to exterior derivatives to p-forms and defined by maps that increase the rank in each wedge product.For example, when These operators d, d, d satisfy the following properties; Moreover, for a p-vector field Ξ (p) and a q-vector field Σ (q) , we have and so on.We next define "inner products" by The other combinations vanish.Given these inner products, L and L are dual vector bundle to each other and L is self-dual.These notions enable us to introduce interior products ι, ι, ῑ as anti-derivatives of d, d, d.Namely we define The other combinations are all zero.They are maps that decrease the rank in each wedge products.They act, for example, like ιξ Note that all the interior products anti-commute with all the exterior derivatives.Then, we have the following properties; These are economically written as where i = (ι, ι, ῑ).For, Here the symbol ˇstands for removing the corresponding part.By using these operations, we define Lie derivatives as Note that these expressions are valid only for the vector spaces that are different from the defining space of the operators.For example, when L X defined by X ∈ Γ(L) acts on ξ ∈ Γ( L), the expression (3.15) is valid.However, when the defining and the acting spaces are the same, we have the following expressions; where [•, •] L , [•, •] L are Lie brackets in L, L, respectively.We note that this rule does not hold in L, namely, we have La 1 a 2 ̸ = [a 1 , a 2 ]L.We also define "Lie-like derivatives" as and a compatibility condition, Here f : M → R is a function on M .We now discuss Lie algebroids in L and L. On the subbundle L, we have the Lie bracket ∂ µ satisfying the Jacobi identity.The same is true for L. Using the operators d, d in (3.3), the anchor maps are generically defined through the following relations ; The simplest case is that the anchor is given by the identity endomorphism ρ = id.Then Here F ∈ Γ((E * ∧ E * ) ⊗ E) is a constant (2, 1)-tensor and E * is the dual vector bundle of E. The symbol ι V for V ∈ Γ(E) stands for the partial contraction using the canonical pairing between E and E * .The twisted bracket must satisfy the Jacobi identity.We can confirm that the condition (3.19) is automatically satisfied when (E, ρ, On the other hand, the Jacobiator for the twisted bracket is calculated as Here, Jac(V 1 , V 2 , V 3 ) is the Jacobiator defined by the ordinary Lie bracket [•, •], which vanishes by definition, and c.p. means the cyclic permutations.For a twisted Lie algebroid, we demand that Jac(V 1 , V 2 , V 3 ) F vanishes at each order in F .This leads to the following conditions; These are constraints on F for the twisted Lie algebroid.
For the L space, we have the triple ( L, ρ, [•, •]L -F ).Here ρ : L → T M 2D+n is the anchor, and we employ ρ = id as in the case of L, L. The twisted bracket associated with [ The conditions in (3.25) are explicitly given by Note that the indices are raised and lowered by the metrices κ αβ , κ αβ .The first line above implies that the flux F α βγ satisfies the Jacobi identity.The second line means for any vector fields * = a 1 , a 2 , a 3 .We stress that this is nothing but the gauge condition in (2.11).

Metric algebroid and extended double
We next reconstruct the twisted C-bracket (2.13) in the geometrical languages.Given the geometrical operations (3.2), (3.3), (3.10), (3.15), (3.17), we find that the twisted C-bracket (2.13) is rewritten as Here the components of vector fields are decomposed as Ξ ).The expression (3.29) is similar to the doubled structure in O(D, D) DFT (see eq. (A.3) in Appendix A) but here it looks rather like a "tripled" structure constructed out of L, L and L. It is therefore tantalizing to clarify the corresponding structure of algebroids associated with the twisted C-bracket (3.29).
In the following, we show that the twisted C-bracket (3.29) in its geometric form defines a metric algebroid in M 2D+n .A metric algebroid is defined by a quadruple (V, [•, •] F , ρ, (•, •)) where V is a vector bundle over a manifold M , [•, •] F is a skew-symmetric bracket, ρ is an anchor, (•, •) is a non-degenerate symmetric bilinear form satisfying the following conditions; Now we have all the ingredients to construct the metric algebroid defined by the twisted Cbracket.Given Lie algebroids (L, ρ, and an anchor ρ = ρ + ρ + ρ : V → T M 2D+n and the bracket (3.29).We also introduce a bilinear form and D = d + d + d.In the following, we show that the quadruple (V, ρ, [•, •] F , (•, •)) defined by this way indeed satisfies the axioms of the metric algebroid.Although the trivial example is the identity map, we leave the anchor maps arbitrary in the following3 .

Proof of Axiom M1
We first study Axiom M1.Namely, for any function f in M 2D+n , we show the following condition is satisfied; The (2D + n)-dimensional vector fields Ξ i (i = 1, 2) are given by Ξ i = X i + ξ i + a i (i = 1, 2).The left-hand side in (3.31) is decomposed as The [a 1 , f a 2 ] F part contains terms other than the Lie bracket; Here [•, •]L -F is the twisted bracket introduced in (3.26).Using the definitions (3.15) and (3.17), we have Note that the interior products anti-commute with the basis of vectors.Then we have where we have used the relation ῑa 1 ( df )a 2 = ρ(a 1 ) • f a 2 and the fact that ( L, ρ, [ (3.38) By using the following relations we therefore obtain Similarly, we have Collecting all together, we find Then we have proved Axiom M1.

Proof of Axiom M2
In order to study Axiom M2, it is convenient to introduce the following quantity; where c.p. is the cyclic permutations.Now we decompose the bracket into the terms in L ⊕ L and L parts; Here [•, •] C is the C-bracket for the ungauged DFT (see (A.3) in Appendix A) and e i = X i + ξ i ∈ Γ(L ⊕ L), (i = 1, 2).Then we have where we have introduced the notation i We have also defined the following quantities; The first term ([e 1 , e 2 ] C , X 3 + ξ 3 ) in (3.45) is evaluated as [6] ( where we have defined Each part in the second term in (3.45) is evaluated as Here we have used the relations Note that the last one follows from the identities; We find that the similar relations also hold for the following quantities, Collecting all together, we find and therefore this results in By summing up the equation (3.57) and the ones with 2 and 3 interchanged, we have Since we have ρ = ρ + ρ + ρ, the third term in the right-hand side in (3.58) becomes The same holds for the fourth term.Then finally we obtain This proves Axiom M2.

Courant algebroid and extended doubled structure
We next study the extended doubled structure for a Courant algebroid defined by the twisted C-bracket (3.29).In the case of ungauged DFT, the C-bracket defines a Courant algebroid provided that the underlying Lie algebroids satisfy the condition of the Lie bialgebroid [18].It has been found that this condition is nothing but the strong constraint in DFT [6].
In the following, we look for conditions that the bracket (3.29) and the Lie algebroids on L, L, L define a Courant algebroid.
Courant algebroid In addition to Axioms M1 and M2 in the metric algebroid, a Courant algebroid is defined by a quadruple (V, ρ, [•, •] F , (•, •)) satisfying the following Axioms; where [•, •] in the right-hand side is the Lie bracket in V.
Proof of Axiom C1 Axiom C1 implies the following relation, where f is an arbitrary function on M 2D+n .The first term in the left-hand side in (3.64) is expanded as On the other hand, the second term in (3.64) is expanded as This does not vanish.Therefore the quadruple (V, [•, •] F , ρ, (•, •)) fails to satisfy Axiom C1 in general.In order (3.67) to be zero, the following conditions are required; We also have the conditions The conditions (3.68) are nothing but the ones that (L, L) becomes a Lie bialgebroid [18].The last condition in (3.69) is a sophisticated expression of the second equation in (3.25).Indeed, when we employ ρ = id, the last equation in (3.69) becomes equivalent to the one in (3.25).
We will comment on (3.68) and (3.69) in due course.

Proof of Axiom C2
We next study Axiom C2.The left-hand side in (3.62) is expanded and rewritten as This again does not vanish in general.In order this to be zero, we have the condition; We note that this condition is automatically satisfied when (3.68) and (3.69) are satisfied.If we keep only the O(D, D) part by dropping out the gauge sector, the skew-symmetric property of the anchor ρ, ρ is naturally recovered [18].We also note that the situation is consistent with the fact that Axiom C1 actually implies Axiom C2 in Courant algebroids [37].
Proof of Axiom C3 Axiom C3 is the modified Jacobi identity for the twisted C-bracket; where T F is defined by (3.56).In the following, we calculate the left-hand side of (3.72), and examine the conditions for Axiom C3.The left-hand side of (3.72) is decomposed as where I ′ 1 , I ′ 2 , I ′ 3 are Γ( L), Γ(L) and Γ( L) parts in Jac F (Ξ 1 , Ξ 2 , Ξ 3 ), respectively.The explicit forms of As is obvious in the right-hand side in (3.56), we expect that the O(F 2 ) terms in Jac F (Ξ 1 , Ξ 2 , Ξ 3 ) must vanish.Indeed, the O(F 2 ) terms in Jac F (Ξ 1 , Ξ 2 , Ξ 3 ) are calculated as This vanishes due to the Jacobi identity of F M N P .We next evaluate the O(F 1 ) terms in Jac F (Ξ 1 , Ξ 2 , Ξ 3 ).Since the I ′ 1 and I ′ 2 parts have essentially the same structures and they interchange by X ↔ ξ, we focus only on I ′ 1 and I ′ 3 parts.Firstly, we have Here [Ξ 1 , Ξ 2 ] C is the C-bracket without the term i Ξ 1 i Ξ 2 F .We have also introduced the notations (3.76) 4 .The first term in the right-hand side in the last expression of (3.75) contributes to DT F while the remaining terms must vanish.This therefore implies the condition; for any quantities * .
The last is the O(F 0 ) terms.We evaluate each term that contains specific quantities.For example, terms that contain one a and two ξs are given by By using the following identity we find I ′ 1 | O(F 0 ),aξξ = 0. Terms that contain one X and two ξs are given by This neither contributes to DT F nor vanishes in general.Terms that contain only a becomes Here we have expanded the Lie(-like) derivatives and collected all the c.p. parts together.Similar treatments have been done for the other parts and we finally obtain the following result; The first, the second and the third terms precisely give dT F | O(F 0 ) .After expanding the fourth to the sixth terms and collecting all together, we find

.83)
In order Axiom C3 holds, we therefore impose the condition η M N ∂ M * ∂ N ξ = 0.In summary, we have the following conditions in I ′ 1 for Axiom C3; Similar calculations in I ′ 2 lead to the conditions Finally, we evaluate the remaining part I ′ 3 .The total expression of I ′ 3 is given by (B.2) in Appendix.The O(F 1 ) in I ′ 3 is evaluated as As in the case of I ′ 1 , we find that the condition F K P Q ∂ K * = 0 is necessary for Axiom C3.We next examine the O(F 0 ) contributions.Since we have many terms in we pick up them one by one.For example, the terms including three as are given by

.87)
Here we have decomposed the Lie(-like) derivatives into the interior products and the derivative operators.It is easy to show that the first and the second lines together with the c.p. parts in the last expression contribute to T F .We find that the terms including two ξs and one a vanish; Here we have used the identity

.89)
We also find that the ξξX terms give a total derivative term; Extended Courant algebroids that accompany gauge structures have been studied in various contexts.In particular, in order to introduce non-Abelian gauge groups, the ordinary generalized tangent bundle is supplemented by the gauge sector.This is obtained, for example, by the generalized reduction of higher dimensional Courant algebroid [40].Mathematically, this is a transitive Courant algebroid [41] which is analyzed in the context of (exceptional) generalized geometry [42].A related concept is the B n -generalized geometry [43] for which the ordinary generalized tangent bundle Here adP G is the adjoint bundle of a principal G-bundle P G over M D for a gauge group G. Applications of these extended Courant algebroids to heterotic theories are discussed in [44,45].The generalized reduction in the mathematical literature has been anticipated in the physical viewpoints [46,47].
As in the case of the The vector field a = a α ∂α ∈ Γ( L) have the index α for the adjoint representation of the gauge group G and we now identify the basis ∂α with the generator T α of the gauge group G.Under these conditions, we find that the twisted C-bracket (3.29) becomes where we have introduced the Lie algebra valued vector fields a i = a iα T α and used the relation Then in the physical spacetime M D , the tangent bundle of the extended doubled space T M 2D+n is identified with the heterotic generalized tangent bundle.

Conclusion and discussions
In this paper, we studied an extended doubled structure of algebroids defined by the twisted C-bracket in the gauged DFT.We first provided the geometrical framework for the twisted C-bracket in the (2D + n)dimensional manifold M 2D+n in which the 2D-dimensional para-Hermitian manifold is embedded.This gives explicit expression of the twisted C-bracket in the Drinfel'd double-like form that is different from the one for the C-bracket in the ordinary DFT.Using the geometric expression of the twisted C-bracket, we showed that the bracket together with the appropriate bilinear form and the anchor maps define a metric algebroid.The fact that this is constructed out of three (twisted) Lie algebroids is an analogue of the Drinfel'd double for Courant algebroids.In order to implement the algebroid structure, we introduced in advance the notion of twisted Lie algebroids for which we find the conditions on the flux F α βγ .They are just the Jacobi identity for F α βγ and the gauge condition F α βγ ∂α * = 0 in the gauged DFT.
We next studied the consistency conditions for Courant algebroids.In addition to the requirement that (L, L) form a Lie bialgebroid [49], we found extra conditions in the gauge sector.These conditions are encoded, in the gauged DFT language, to the strong constraint and the gauge condition.The situation is similar to the Drinfel'd double for Courant algebroids in the ungauged DFT but here it is extended to include the gauge sector.When the conditions for the Courant algebroid are imposed, we found that the twisted C-bracket reduces to the Courant bracket in the heterotic theory.We showed that this is associated with the heterotic version of the natural isomorphism which allows us to identify the tangent bundle of the extended doubled space M 2D+n with the heterotic generalized tangent bundle.This implies that the heterotic Courant algebroid is decomposed into three pieces of algebroids.
In our analysis, it was shown that even for algebroids whose dimensions are not even number, there could be doubled-like structures.This implies that the gauged supergravities itself exhibit doubled-like structure behind them [50].The Drinfel'd double-like structures in gauged DFT will be useful to analyze T-dualities among heterotic supergravity solutions and gauged supergravities.Relations to L ∞ -algebra are also interesting issues [9,[51][52][53].It would be interesting to find a global structure of the gauge symmetry.For example, trying to the "coquecigrue problem" [54][55][56] i.e. finding the integrated, group-like structure corresponding to the twisted C-bracket is an interesting direction.We will come back to these issues in future studies.
valuable feedback and insightful comments on this manuscript.Their meticulous review process has greatly contributed to improving the quality and accuracy of the paper.The work of S. S. is supported in part by Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number JP20K03952.The work of H. M. is supported by Grant-in-Aid for JSPS Research Fellow, JSPS KAKENHI Grant Numbers JP22J14419 and JP22KJ2651.
) where B µν is a D × D anti-symmetric matrix.The indices M, N, . . .= 1, . . ., 2D + n are raised and lowered by the O(D, D + n) invariant metric; e B v E b e g B N M i 3 x A b p H P o E 7 s Y c 5 8 Q 7 4 n D x u 9 x l e N r x s P O O v 5 c 0 L m F Z L 7 N L 7 9 D 4 k q I v A = < / l a t e x i t > x µ

Figure 1 :
Figure 1: A schematic picture of the tri-foliation.Each sheet represents M 2D specified by xα = const.The red and blue lines indicate spaces given by xµ = const.and x µ = const.

1 2 (L a 1 a 2 − L a 2 a 1 ) = tr a 1 da 2 − a 2 da 1 and ῑa 2 ῑa 1 F
= a 1 , a 2 by using the relation T α , T β = −F γ αβ T γ for the Lie algebra bracket •, • and the normalization trT α T β = 1 2 κ αβ .The bracket (4.3) is nothing but the heterotic Courant bracket on T M D ⊕ adP G ⊕ T * M D[48].This fact makes us to find a heterotic version of the natural isomorphism[14,16];T M 2D+n ≃ T M D ⊕ adP G ⊕ T * M D .(4.4) .17) They are valid in any cases discussed above.Lie algebroids on L, L We next introduce Lie algebroid structures in each subbundle.
A Lie algebroid is a triple (E, ρ, [•, •])where E → M is a vector bundle over a base manifold M , ρ : E → T M is an anchor map, and [•, •] is the Lie bracket on E satisfying the Jacobi identity