Exceptional Algebroids and Type IIA Superstrings

We study exceptional algebroids in the context of warped compactifications of type IIA string theory down to n dimensions, with n≤6$n\le 6$ . In contrast to the M‐theory and type IIB case, the relevant algebroids are no longer exact, and their locali moduli space is no longer trivial, but has 5 distinct points. This relates to two possible scalar deformations of the IIA theory. The proof of the local classification shows that, in addition to these scalar deformations, one can twist the bracket using a pair of 1‐forms, a 2‐form, a 3‐form, and a 4‐form. Furthermore, we use the analysis to translate the classification of Leibniz parallelisable spaces (corresponding to maximally supersymmetric consistent truncations) into a tractable algebraic problem. We finish with a discussion of the Poisson–Lie U‐duality and examples given by tori and spheres in 2, 3, and 4 dimensions.


DOI: 10.1002/prop.202200027
M-theory setup, while the latter was devoted to the type IIB case. In both cases, a local classification result was proved, a method for constructing Leibniz parallelisable spaces [22] was provided, and a general notion of the Poisson-Lie Uduality was studied, extending the exceptional Drinfeld algebra construction of [25,30].
Although quite distinct as far as the technical details are concerned, the Mtheory and type IIB cases nonetheless have one thing in common -they both correspond to a certain "minimal" class of exceptional algebroids, called exact. The purpose of the present text is to complete the "triangle" and provide a detailed derivation of the analogous results in the type IIA case. In more detail, we describe the relevant subclass of exceptional algebroids, called type IIA algebroids and prove that they locally correspond to a type IIA version of the exceptional tangent bundle [4,5,11,12,17,19] (Subsection 3.5 and Theorem 4.1). We then study the relation between embedding tensors and Leibniz parallelisations. We show which embedding tensors define such a parallelisation -they correspond to a pair of an elgebra (exceptional algebroid over a point) together with a suitable coisotropic subalgebra, satisfying some mild conditions (Theorem 5.1). This result gives a simplification and a slight refinement of the result of Inverso. [20] We then describe the Poisson-Lie U-duality phenomenon and its compatibility with the supergravity equations. We finish by discussing examples of Leibniz parallelisations over tori, S 2 , S 3 , and S 4 , and we explain how the ordinary U-duality and generalised Yang-Baxter deformations fit in our framework.
In addition to being more technically involved, the type IIA setup exhibits an important difference from the M-theory and type IIB case -the local moduli space of the type IIA exceptional algebroids is not trivial, but consists of 5 points (see the picture in Theorem 4.1). This corresponds to two deformations of the type IIA theory. [18,29] Also, similarly to the IIB case, the "algebraic calculation" of the possible twists of the bracket reveals the option of having a non-physical twist by a vector field. This complication results in an extra trace condition in Theorem 5.1, which is not present in the M-theory case [7] (but it appears in the type IIB case [8] ).

Exceptional Algebras
In order to define exceptional algebroids, we first discuss the neccessary algebraic prerequisites, following. [7] The central role will Here E 6(6) is the connected and simply connected Lie group whose Lie algebra 6(6) is the split real form of the exceptional complex Lie algebra of rank 6. The cases with n = 5, 4, and 3 also correspond to split real forms of complex Lie algebras which, although not exceptional, belong to a certain natural generalisation of the exceptional family, c.f. their Dynkin diagrams below.
For each of these groups, there are two representations, labeled E and N, which will be of particular interest to us. We list these below the Dynkin diagrams of the (semisimple part of) E n(n) . Both E and N correspond to fundamental or trivial representations of (the simple factors of) E n(n) . The nodes corresponding to E and N are marked by black and blue colour, respectively. 2 In fact, we will be interested in the slightly larger group G := E n(n) × ℝ + . We will consider E and N as G-representations, by giving them ℝ + -weights 1 and 2, respectively.
Crucially, we have two symmetric G-equivariant maps satisfying the following property: First, taking the dual of the second map we get a map N → E ⊗ E. Let now ′ : where is seen as a subalgebra of E * ⊗ E.
To avoid complicated notation, we will not give the maps from (1) specific names, but will instead refer to them using a subscript (signifying a projection). For instance the image of u ⊗ v ∈ E ⊗ E under the first map will be denoted simply by (u ⊗ v) N . We will use the same type of notation also when dealing with (partial) duals of these maps. For instance, the image of One should think of the maps (1) as a generalisation of the ordinary inner product, which would correspond to taking the group O(p, q) with E and N being the vector and scalar representations, respectively. This analogy also motivates the following definitions.
We say a subspace V ⊂ E is Note that a more unified description can be obtained if we decompose everything in terms of a subalgebra (n, ℝ) ⊂ n(n) ⊕ ℝ, as will be shown in Subsection 2.2.
Note that coisotropic spaces can be equivalently characterised by the property (V • ⊗ N) E ⊂ V, see [7]. 3 Finally, note that for any n > 2, we have the relation where = − dim E 9−n . For clarity and later reference, we give a table of the corresponding values:

M-theoretic Decomposition
In order to be more explicit, let us perform the decomposition under a Lie subalgebra (n, ℝ) ⊂ . We then obtain The first decomposition is arranged such that (T) acts in the standard way on all the summands, while ℝ acts with weights 0, 1, 2 on , E, N, respectively. Denoting the forms and multivectors in the first line by a and w (with subscripts indicating the degree of the form/multivector), and an element of E by u = X + 2 + 5 , the remaining parts of the action of on E are The nonzero bits of the symmetric map E ⊗ E → N are given by The map E * ⊗ E * → N * is given by precisely analogous formulas, up to an (unimportant) overall factor which is fixed by the condition (2), see [7]. In other words, using a suitable inner product on T to identify E ≅ E * and N ≅ N * , the two maps (1) coincide. In particular, there is a bijection 4 between the possible isotropic subspaces of dimension k and coisotropic subspaces of codimension k. 3 We remark that this definition of isotropy can be seen as a distant relative of the isotropy studied in the context of higher Dirac structures, [32] which is defined for a different pair of E, N. 4 depending on the choice of the inner product www.advancedsciencenews.com www.fp-journal.org

Structure of Isotropic Subspaces
The space of possible isotropic subspaces has an interesting structure, which can be captured by the following Hasse diagram (drawn for all n ∈ {2, … , 6}). This is to be read as follows: The nodes in the i-th line (counted from the bottom) represent isotropic subspaces of dimension i, up to an action of an element of G. If an isotropic subspace is a subspace of a larger isotropic subspace, the corresponding nodes are linked by an up-going line. Black nodes correspond to maximally isotropic subspaces. Because of the above bijection, these diagrams also capture the structure of coisotropic subspaces.
We see that there exists a unique (up to the G-action) ndimensional isotropic subspace, i.e. n-dimensional isotropic subspaces form a single orbit of G. In terms of the M-theoretic decomposition, this corresponds to T ⊂ E. Correspondingly, coisotropic subspaces of codimension n form a single G-orbitthey are all equivalent to ∧ 2 T * ⊕ ∧ 5 T * ⊂ E. Any coisotropic subspace from this orbit will be called type M.
In contrast, n − 1-dimensional isotropic subspaces form two G-orbits. One of this is maximal and the other one is not (it can be enlarged to an n-dimensional isotropic subspace). This can be most readily seen in the n = 2 case, where we have E = T ⊕ ∧ 2 T * , with T = ℝ 2 : one-dimensional isotropic subspaces are either given by any 1-dimensional subspace of T, or by the 1dimensional space ∧ 2 T * (the former can be enlarged to T).
Correspondingly, coisotropic subspaces of codimension n − 1 are of two types: • the ones which do not contain any smaller coisotropic subspace, called type IIB (depicted by a black node) • those that do contain a smaller coisotropic subspace, called type IIA (depicted by a grey node).
Note that one can think of the above Hasse diagrams as capturing the structure of toroidal compactifications of type IIA/B string theory and M-theory, or as capturing the structure of maximally supersymmetric theories in various dimensions. For instance, they show that reducing the type IIA and IIB string theories on a single circle leads to the same theory.
Exceptional algebroids corresponding to the type M and IIB cases were discussed in [7] and [8], respectively. In this paper, we concentrate on the third case, given by type IIA.

Type IIA Subspaces
In terms of the M-theoretic decomposition, any type IIA subspace is equivalent (i.e. can be related by a G-transformation) to the where L is any 1-dimensional subspace of E, spanned by some vector e ∈ T.
Furthermore, it will be useful to consider the following Lie algebra A simple calculation shows that with T * ⊗ L ⊂ (T). Note that ℝ ′ ⊂ acts on T, ∧ 2 T * , ∧ 5 T * with weights 0, 1, 2, respectively. Perhaps more explicitly, choosing a decomposition T = L ⊕  , with  ≅ ℝ n−1 , we get a subalgebra ( ) ⊂ (T), under which as follows immediately from the M-theoretic decomposition. The type IIA subspace W then corresponds to while its complement  is isotropic. One can write down a similar decomposition for N and . However, we will not do so -when doing "algebraic" calculations, we will instead work with the M-theoretic decomposition of E (with a chosen subspace L).
Finally, we note an important equivalent characterisation of type IIA subspaces: W ⊂ E is of type IIA if and only ifŴ := (W • ⊗ N) E is a subspace of W of codimension 1. This follows from a simple case-to-case check using the classification of coisotropic subspaces. Explicitly, taking the identification (6), we havê i.e.Ŵ misses the ∧ 0  * -part.
Remark. The existence of the other inequivalent coisotropic subspace of codimension n − 1 is reflected in the existence of another inequivalent embedding of ( ) into , under which we get with S := ℝ 2 . In this case (S ⊗  * ) ⊕ ∧ 3  * ⊕ (S ⊗ ∧ 5  * ) is a type IIB subspace. We will return to this briefly in Theorem 3.1.

Definition
We now proceed to the definition of exceptional algebroids (or simply elgebroids). [7] For simplicity we will use the same letters to denote G-representations and the corresponding associated vector bundles. For instance, the crucial object will be a vector bundle E → M, which is assumed to transform in the representation E of the group G. Having such a bundle, we then automatically have another vector bundle N → M (it is associated to the same principal G-bundle as E).
By definition, an exceptional algebroid (or elgebroid) is given by the data of a vector bundle E → M (transforming in the corresponding G-representation), together with an ℝ-bilinear bracket called the anchor, and an ℝ-linear map [a, ⋅ ] preserves the G-structure, wheredf : Remark. Following an approach from [31] (in the Courant algebroid case), the axiom (8) implies that any a ∈ Γ(E) induces a vector field s a on the total space E such that Axiom (11) then says that this vector field, when lifted to the frame bundle of E, preserves the subbundle given by the Gstructure. Equivalently, if e is a local G-frame of E, then so is e ′ := e + [a, e ], up to the first order in (for any a).
Note that exceptional algebroids form a particular subclass of vector bundle twisted Courant algebroids of [15].

First Consequences of the Definition
First, note that since E ⊗ E → N is surjective,  is uniquely determined in terms of the other data (the bundle, G-structure, bracket, and anchor). Furthermore, symmetrising (7) in a and b Second, from the axioms (7) and (8) one easily shows that the anchor intertwines the bracket on E and the commutator of vector fields: Applying to (9) we then get In conjunction with axiom (10) this implies In other words, setting we get a complex As shown in [7], this is equivalent to saying that ker ⊂ E is coisotropic at every point.

Exact Elgebroids
We say that the elgebroid is exact if (14) is an exact sequence, i.e. if the homologies H 1 and H 2 vanish. In this case we have the following classification result. [7,8] Theorem 3.1. Exact elgebroids have locally one of the two following forms. Either www.advancedsciencenews.com www.fp-journal.org In both cases the anchor map is given by the projection onto the first factor. In the second case, the arrow signifies the corresponding tensor is valued in S = ℝ 2 and we used  M for the tangent bundle over an n − 1-dimensional base manifold.

Type IIA Elgebroids
We will be interested here in the third case, corresponding to the type IIA string. This requires that the elgebroid is non-exact, in a certain minimal sense. More precisely, we will say that an elgebroid is of type IIA if the following two conditions hold at every point on M: The second condition says simply that is surjective. Setting W := ker , the first condition is equivalent to saying that (W • ⊗ N) E ⊂ W is a subspace of codimension 1 at every point. (We use the fact that W • = im T .) Following the discussion from Subsection 2.4 we thus conclude: An elgebroid is of type IIA iff is surjective and ker is of type IIA at every point.

Type IIA Exceptional Tangent Bundle
The prime example of a type IIA elgebroid is the type IIA exceptional tangent bundle. This is determined by an n − 1-dimensional manifold M together with locally constant functions 0 and satisfying 0 = 0 (if M is connected, we have that either 0 = 0 and is constant, or the other way round). We will use  M to denote the tangent bundle, in order to distinguish the type II and M-theory cases.
We then take the bundle and the anchor is given by the projection onto the first summand.
Note that, just as the type IIA theory can be obtained by a Kaluza-Klein reduction of M-theory, we can obtain (a class of) type IIA exceptional tangent bundles from certain M-theory ones. For instance, if the base manifold of an M-theory exceptional tangent bundle is a product M × S 1 , then restricting to the S 1invariant vectors and forms gives a type IIA tangent bundle over M. However, not all type IIA exceptional tangent bundles arise this way -for instance the above 0 -deformation cannot be obtained by such a reduction (the other parameter can be obtained from a twist of the M-theory bracket by a 1-form, see [7]). A similar type of reduction was studied in [6,30] in the case of Leibniz parallelisations (see Subsection 5.1).

Classification
Our first goal is to prove the following theorem

Strategy of the Proof
We follow the strategy laid out in [7,8] for the M-theory and type IIB case. In particular, it turns out to be convenient (also for the next Section) to first study a weaker object, called a type IIA preelgebroid. This is given by the same axioms (including dim H 1 = 1 and dim H 2 = 0), except that the Jacobi identity (7) is replaced by the condition (13). We then proceed as follows.
(1) We use the axioms to constrain the form of the type IIA preelgebroid. This will be locally parametrised by a set of tensors, which will essentially correspond to the twists. (2) We use the Jacobi identity to derive a set of algebraic and differential (Bianchi) identities for the twists. (3) During these procedures, some choices will be required. A change in these choices can be interpreted as a gauge transformation for the twists. We use this freedom to locally remove the twists. Some remnants of the scalar twists will remain and will correspond to the parameters 0 and .

Bundle
Assume E is a type IIA pre-elgebroid. First, note that we can locally make the identification with the anchor being simply the projection onto the first factor. This follows by the same argument used in [7,8], based on the facts that • The identification (5) provides a decomposition of E into a direct sum of an isotropic subspace  and a type IIA subspace  (15) is not unique. Two such identifications differ by an anchor-preserving G-transformation, i.e. by a transformation belonging to the subgroup N ⊂ G, with Lie algebra . This will be interpreted as a gauge transformation.

Bracket
Choose local coordinates x i on M. This leads to a trivialisation of E, where now we take again  := ℝ n−1 . In other words, and we have a well defined action of vector fields and of the differential d on Γ(E) -they leave the  ⊕  * ⊕ (∧ 0  * ⊕ ∧ 2  * ⊕ ∧ 4  * ) ⊕ ∧ 5  * -part intact and only act on C ∞ (M).
Using this trivialisation, from the axioms (8)-(10) it follows that is C ∞ (M)-linear in both a and b. In other words, we can write where A : E → End(E) is a tensor. Furthermore, the axiom (11) ensures that in fact at every point of M we can see A as a map of two vector spaces, In fact, we can make an even stronger constraint based on the following observation.
Similarly, axiom (10) implies that where B : N → E is again C ∞ (M)-linear. Finally, taking a, b constant in (9) gives

Algebraic Part of the Calculation
To simplify the problem a bit, we shall use the M-theoretic description and formulas. In other words, we define the bundle TM :=  M ⊕ L, where L is an auxiliary product line bundle, spanned by a section e (c.f. Subsection 2.4). In particular, we have again though we have to keep in mind that (despite the notation) TM is now not the tangent bundle of M. A straightforward calculation (see Appendix A) shows that (17) and (18) imply for a set of twists This corresponds, in terms of vectors and differential forms on M, to a pair of scalars, a pair of 1-forms, a 2-form, a 3-form, a 4-form, and a vector.

Explicit form of the Bracket
In order to be able to use the differential geometric formulas on TM (instead of  M), we replace the space M by M × ℝ, with the coordinate y on ℝ. At every point, the tangent bundle of this space coincides with TM, where we identify y and e. For simplicity, we will drop the index y and write just .
On the other hand, as the y-direction is non-physical, the coefficient functions X i and i…j will be taken to be independent Fortschr. Phys. 2022, 70, 2200027 www.advancedsciencenews.com www.fp-journal.org of y. Thus we have for instance X = x + s 0 , 2 = s 2 + s 1 ∧ dy, 5 = s 5 + s 4 ∧ dy, where x + s 1 + s 0 + s 2 + s 4 + s 5 can be seen as a section of (15). Sticking to this notation, the bracket determined in the previous subsections is The terms with no twists arise simply from writing explicitly the first two terms on the RHS of (16) (e.g. using the explicit formulas for from [11]). Note that when acting on vectors and forms whose components are independent of y, the operation  f acts tensorially (without derivatives) -at every point we have

Jacobi and Bianchi
Requiring now the Jacobi identity for the bracket (19), one gets constraints for the twists. These consist of the constraint = for some function , and the requirement that the following expressions vanish We will see that the latter set of equations gives rise to the Bianchi identities for the twists. In fact, these constraints/vanishings follow already from the vanishing of [n 1 , ⋅ ] for n 1 ∈ T * ⊂ N (which is a simple consequence of the Jacobi identity, see Subsection (3.2)). 6 For instance, taking n 1 = fdy and keeping only the ℝ ′ -part of the expression, we are left with where the second term contains no derivatives of f . Thus its vanishing requires = for some . The rest of the constraints is obtained in a similar fashion.
Taking into account that = , the bracket takes the following simpler form -we shall refer to it as the twisted bracket.
. 6 A curious exception is the case n = 5, where one does not obtain the Bianchi identity for F 4 in this way. To derive this Bianchi identity one can instead use the Jacobi identity with a, b, c being vectors.
An explicit form of this bracket in terms of geometric structures and operations on M can be found, together with the relevant Bianchi identities (see below), in Appendix B.

Gauge Transformations
Infinitesimal gauge transformations are parametrised by an element ∈ C ∞ (M) ⊗ , which we write (using (4)) as = 0 + + 3 + 6 , witĥ:= 1 ⊗ . Assuming the above constrained form of , and writing [ ⋅ , ⋅ ]  for the twisted bracket with  = ( , F 1 , F 2 , F 4 ), we have from which we can read off the following gauge transformations: 7 The remaining non-displayed gauge transformations (including the action of 6 ) are all trivial.

Type IIA Language
Decomposing the twists as the expressions from the beginning of Subsection 4.6 reduce down to the vanishing of This provides a set of equations for the twists The first one, 0 = 0, is purely algebraic, while the rest gives differential Bianchi identities. Concerning the gauge transformations, writing 7 To determine the expressions, it is enough to examine the transformation properties for [X, Y], for the vector part of [ 2 , X], and for [ 2 , X] with i 2 = 0. These will determine transformations of (F 2 , F 4 ), , and F 1 , respectively. www.advancedsciencenews.com www.fp-journal.org we get Remark. The non-zero-form twists can be associated with field strengths of the bosonic fields appearing in the restriction of the type IIA supergravity to n − 1 dimensions, with a warp factor. This consists of • two scalars -the dilaton and the warp factor • a 1-form and a 3-form (Ramond-Ramond fields) • a 2-form (Kalb-Ramond field).

Removing the Twists
We now use the gauge transformations to remove most of the twists one by one, taking account of the relevant Bianchi identities. The equation 0 = 0 implies that we can (locally) distinguish two cases -either = 0 or 0 = 0.
Suppose first that = 0. We notice that 1 is closed. We can thus locally use a (finite version) of the b 0 -transformation to set 1 to zero. We will write this step as b 0 ⇝ 1 . The Bianchi identities then reduce to the vanishing of while the remaining gauge transformations take the form a 0 1 = −da 0 , a 0 3 = a 0 3 , a 0 4 = a 0 4 , Since the Bianchi identity for 2 was now simplified to d 2 = 0, we can use an appropriate a 1 -transformation to set 2 = 0, i.e. a 1 ⇝ 2 . Crucially, note that the a 1 -transformation will not generate a nonzero value of the twists which were already set to zero (in our case 1 and ). Continuing this procedure, we can now perform and we are left with only one non-zero twist 0 , which is (locally) constant. We now note that we can still scale 0 using a constant b 0 -transformation, and so we can set it to one of the three values: −1, 0, 1. Suppose now that 0 = 0. Similarly to above, after performing we end up with a single locally constant twist , which can again be scaled via a b 0 -transformation to one of the values −1, 0, 1. Writing the bracket in terms of operations on the manifold M (instead of M × ℝ), we obtain precisely the bracket of the type IIA exceptional tangent bundle. This concludes the proof of the Theorem.

Embedding Tensors and Leibniz Parallelisations
Out second main goal is to study the relation between embedding tensors and Leibniz parallelisations. Again, we follow here the general approach from [7,8], where the M-theory and type IIB cases were discussed.

Definitions
A type IIA elgebroid E ′ over M ′ is said to be Leibniz parallelisable if there exists a global G-frame e such that [e , e ] = c e , with constant structure coefficients c . Such a choice of frame provides a Leibniz parallelisation of the elgebroid. We shall restrict our attention to manifolds M ′ which are compact and connected.
Given such a parallelisation, we have in particular that E ′ is a product bundle, E ′ ≅ M ′ × E, where the vector space E lies in the corresponding representation of G. Elements of E correspond to constant sections of E ′ . Since the structure coefficients are constant, E inherits a well-defined bracket. Understanding E as a vector bundle over a point, this defines an exceptional algebroid structure on E (with = 0), i.e. E is an elgebra (see Subsection 3.1).
It was shown in [22] that Leibniz parallelisations correspond to consistent truncations to theories with maximal supersymmetry. The associated elgebras then correspond to the embedding tensors of these lower-dimensional theories.
We now answer the following landscape-type question: Which elgebras (embedding tensors) correspond to Leibniz parallelisations?

Actions of Elgebras
Assume we have a Leibniz parallelisation over a (compact connected) manifold M ′ . First, note that (as a consequence of the axioms) it can be reconstructed from the following data:  = ((m, x)) for any m ∈ M ′ . Because of (13), is a map of algebras. In analogy with the case of ordinary Lie algebras, we will thus say that is www.advancedsciencenews.com www.fp-journal.org an action of E on M ′ . We conclude that any Leibniz parallelisation gives rise to an action of an elgebra on a manifold M ′ . Conversely, this action determines the parallelisation.

From Elgebras to Lie Algebras
Recall that a general elgebra E is not a Lie algebra, since the bracket is not necessarily skew-symmetric. However, the subspace is a (both-sided) ideal. This follows from (12) and the fact that Consequently, the quotient E := E∕I inherits a skew-symmetric bracket (satisfying Jacobi identity), i.e. it is a Lie algebra.
Note that the projection gives a bijection between Lie subalgebras of E and subalgebras 8 of E containing I. The Lie subalgebra of E corresponding to a subalgebra V ⊂ E will be denoted by V . 9 Finally, any action of an elgebra E on a manifold induces a corresponding action of E , since If the action was transitive ( is surjective at every tangent space on M ′ ), so is the action of E .

From Manifolds to Algebras
On an exact elgebroid the anchor map is surjective. Thus any Leibniz parallelisation induces a transitive action of a Lie algebra E on M ′ . Since we assume M ′ to be compact and connected, this in turn implies that M ′ ≅ G E ∕H, where G E is the connected and simply-connected Lie group integrating E and H ⊂ G E is a Lie subgroup with a subalgebra ⊂ E .
Using the correspondence between subalgebras of E and E, we have = V for some subalgebra V ⊂ E containing I. Thus a Leibniz parallelisation gives rise to a pair (E, V), with I ⊂ V. Furthermore, since ker ′ is, at the coset of 1 ∈ G E , identified with V, we see that V has to be a type IIA subspace of E.
Conversely, we can ask: Which pairs (E, V) of an elgebra and its subalgebra, with V being a type IIA subspace containing im , correspond to a Leibniz parallelisation?
This is answered by the following Theorem. A similar result was derived, using different methods, in [20].

Theorem 5.1. Let n > 2. Suppose E is an elgebra and V ⊂ E a subalgebra such that im  ⊂ V and V is a type IIA subspace, with G V ⊂ G E a closed subgroup. Then the pair (E, V) defines a Leibniz parallelisation (over G E ∕G V ) if and only if the following condition holds:
Tr E ad a = −1 Tr V ad a ∀a ∈V, Remark. We assume here that G E is the connected and simply connected group with the Lie algebra E . Note that we have, in general, some freedom in the choice of the subgroup G V ⊂ G E corresponding to a given V . These differ in the number of connected components. For instance, for V = 0 and G E = SU (2) we can take G V = {1} or G V = {±1}, leading to the quotients SU(2) ≅ S 3 or SO(3) ≅ ℝℙ 3 , respectively.

Proof of the Theorem
Suppose E is an elgebra and V ⊂ E is a subalgebra which contains im  and is of type IIA. Supposing the corresponding type IIA elgebroid exists, it necessarily has the form with the anchor and bracket given by the following formulae. First, where [g] ∈ G E ∕G V denotes the coset of g ∈ G E . (In other words, the anchor is the infinitesimal action of G E on G E ∕G V .) The bracket is where [ ⋅ , ⋅ ] is the bracket on E and we have used the identifica- A simple check reveals that this object always satisfies the axioms of a type IIA pre-elgebroid. It remains to check under what conditions the Jacobi identity is satisfied -in other words, when the Jacobiator vanishes. We first note that the Jacobiator of constant sections does vanish, as a consequence of the Jacobi identity of E. Consequently, the Jacobiator vanishes completely iff it is a tensor (i.e. C ∞ (M ′ )-linear in all slots).
We know that any type IIA pre-elgebroid has locally the form (19), for some general , F 1 , F 2 , F 4 . (For a general pre-elgebroid the twists are not constrained by the Bianchi identities.) A straightforward calculation shows that for such bracket the Jacobiator is a tensor iff = for some function . Let nowK := ((ker ′ ) • ⊗ N ′ ) E ′ be the codimension-one subspace of ker ′ (see Subsection 2.4). This corresponds to the www.advancedsciencenews.com www.fp-journal.org It is now easy to see that the condition = (for some ) corresponds precisely to the following condition: Note that sinceK ⊂ ker ′ , for a ∈ Γ(K) we have [a, fb] ′ = f [a, b] ′ and so [a, ⋅ ] ′ has a meaningful (pointwise) trace. Using (22) and (3), the above trace condition is equivalent to Tr E [a [g] , ⋅ ] = Tr E (da) [g] , ∀a ∈ Γ(K), ∀g ∈ G E .
Finally, using (21) and relating the expression to the corresponding expression at g = 1, this is equivalent to (see [8] for a detailed derivation) Using Tr E∕V ad a = Tr E ad a − Tr V ad a , we obtain the condition from the statement of the Theorem, which concludes the proof.

Poisson-Lie U-duality and Examples
Let us now discuss the Poisson-Lie U-duality phenomenon and provide some examples. Some details can be found also in [13].

Poisson-Lie U-duality
Assume we have an elgebra E which admits multiple type IIA subalgebras V, which contain im  and satisfy the trace condition. Then the corresponding Leibniz parallelisable spaces are said to be Poisson-Lie U-dual. Poisson-Lie U-duality is a phenomenon introduced in the works [25,30] and was given an interpretation in terms of elgebroids in [7]. It generalises both Uduality and Poisson-Lie T-duality of Klimčík and Ševera. [21] To see one of the implications, let us briefly comment on an aspect of the corresponding supergravity.
Taking a warped compactification of the type IIA supergravity down to n-dimensions, we obtain a supergravity which can be succinctly described via a type IIA elgebroid. Ignoring for now the global issues, this essentially corresponds to the exceptional tangent bundle of type IIA. The bosonic degrees of freedom correspond to the reduction of the structure group from G to its maximal compact subgroup K(G) and the dynamics is encoded in the vanishing of a suitable curvature tensor, constructed using (generalised) torsion-free K(G)-compatible connections, 10 see [11] and the references within.
The formulation of the theory in terms of elgebroids immediately implies the compatibility of the supergravity equations of motion with the Poisson-Lie U-duality. This was shown in [7] in the M-theory case -however, the same result applies also to the type IIA (and type IIB) case. This is because in the type IIA (and also IIB) case one uses exactly the same expressions for the curvature tensors and the same reduction of the structure group as for the M-theory case, see [11] (the only change is that in type II we decompose the representations under the ( )-subalgebra of , instead of (T) ⊂ ).

Torus
We start with the simplest example, taking the elgebra with In this particular example, let us abandon the requirement that G E is simply connected and take it to be the torus T dim E . In that case, if we take any type IIA subspace V ⊂ E, which corresponds to a closed subgroup of G E , we obtain a Leibniz parallelisation on the quotient T n . Different choices of the type IIA subspace are linked by the action of the exceptional group -they correspond to different dual tori. This is the standard U-duality.

2-sphere
Take n = 3. A particularly nice example of an elgebra corresponds to a Leibniz parallelisation of the 2-sphere, discussed in [14]. It can be elegantly obtained from the M-theoretic parallelisation of the group SU(2), using the frame of left-invariant tensors. This has the following form: where ad denotes the (co)adjoint action on (2) and ∧ 2 (2) * , and is the Chevalley-Eilenberg differential. The subspace ∧ 2 (2) * is of type M and it coincides with the ideal I. Consequently, taking a (1)-subalgebra of (2), the subspace V = (1) ⊕ ∧ 2 (2) * is of type IIA and contains I. As one easily sees, it is also a subalgebra and satisfies the trace condition. Consequently, we get a Leibniz parallelisation on the manifold

3-Sphere
This corresponds to n = 4. Since the 3-sphere can be identified with the group SU(2), it automatically admits a Leibniz parallelisation. The elgebra is www.advancedsciencenews.com www.fp-journal.org for any 0 , ∈ ℝ. The corresponding type IIA subalgebra is of the form
In particular, we have im  = V 5 ⊂ E. Any subalgebra V ⊂ E of codimension 4, which contains im , is necessarily of the form for some subalgebra (4) ⊂ (5). It is easy to see that this is coisotropic: First, identify 10 ≅ ∧ 2 ℝ 5 , 5 ≅ ∧ 4 ℝ 5 , and 1 ≅ ∧ 0 ℝ 5 . Taking the standard inner product on ℝ 5 , we get an identification E * ≅ E. Then the map E * ⊗ E * → N * , restricted to 10, becomes For the above V, we have V • = e ∧ ℝ 5 ⊂ ∧ 2 ℝ 5 , for some vector e ∈ ℝ 5 . This clearly satisfies the coisotropy condition. Furthermore, consider the subspace (4) ⊕ V 5 ⊂ V. Its annihilator is identified with V • ⊕ 1. Since any map from 1 ⊗ 10 or 1 ⊗ 1 into N = 5 ⊕ 5 is automatically zero, we get and so (4) ⊕ V 5 is coisotropic (and hence a type M subspace). Thus V is a type IIA subspace.
It also satisfies the trace condition and hence leads to a Leibniz parallelisation of the space 11 This example extends to the more general case, where we replace the Lie algebra (2) by an arbitrary Lie algebra of dimension n ∈ {3, … , 6}. Nevetheless, (2) stands out as the only compact simple Lie algebra in this dimension range.

Conclusions
We have shown how the Leibniz algebroids appearing in the study of type IIA string theory fit naturally in the framework of exceptional algebroids. We have proved a classification result which shows that type IIA exceptional bundles admit an abstract characterisation (in terms of a natural asociated sequence). This in especially convenient for the purposes of Poisson-Lie U-duality -in particular we were lead to an elegant and concise description of this duality as well as to the proof of its compatibility with the supergravity equations of motion.
We have also simplified and refined slightly the result of Inverso [20] concerning the correspondence between embedding tensors and Leibniz parallelisations. This leads to an efficient algorithm for searching for maximally supersymmetric consistent truncations. Although we do not attempt to provide a complete classification of such truncations here, we showed how the known sphere examples from the literature fit into our framework.
From a more mathematical perspective, this article might prove to be of interest due to the fact that -even though they provide a natural extension of exact Courant algebroids -the type IIA exceptional algebroids have a nontrivial (local) moduli space, corresponding to two possible deformations of the type IIA theory.
Similarly to [7,8], the present work focused solely on maximally supersymmetric consistent truncations. However, the authors believe that similar technology can be applied also to the non-maximally supersymmetric case, where relatively little is known and more systematic theory can help to find new consistent truncations. These investigations are left for a future work.

Appendix A: The Calculation
Let us write A = A 0 + (A 1 ⊗ e) + A 3 + A 6 for a map A : E → . We now need to solve the equation for some map B = B 1 + B 2 + B 5 : N → E.
Due to the fact that decomposable 2-forms span the whole ∧ 2 T * , this result is valid for an arbitrary 2-form 2 .
Putting things together,