Adjoint orbits, generalised parallelisable spaces and consistent truncations

The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an explicit expression for a generalised frame is given in the case when the orbits are regular, but subtleties arise when they become degenerate. In the case of regular orbits, the resulting space is a globally flat fiber bundle over $\mathcal{O}$ which can be made compact, allowing for a generalised Scherk-Schwartz reduction. This means these spaces should admit consistent supergravity truncations. For degenerate orbits, the procedure hinges on the existence of a suitable metric, allowing for a consistent normalisation of the generalised frame.


Introduction
The story of consistent truncations is a long one within supergravity, and classifying all possible consistent truncations remains an open problem. The classical examples of Scherk-Schwartz reductions [1,2] on local group manifolds [3,4], together with the exceptional consistent truncations on spheres [5][6][7][8][9] were unified as generalised Scherk-Schwartz reductions 1 in [19] using the language of generalised geometry.
The well-known result of local group manifolds critically hinges on them being (Liebniz) parallelisable. Using generalised geometry, this requirement can be relaxed to generalised Leibniz parallelisability of the space [20], which allows for consistent truncations on certain homogenous spaces, such as spheres [19], twisted tori and hyperbeloids [21,22] 2 . These results makes the question of which manifolds admit a generalised Leibniz parallelisation an interesting one. In [28], a first step towards a systematic construction of these spaces, and the generalised frames on them, was taken using exceptional field theory [29][30][31][32][33][34][35][36]. The aim of this note is to further contribute to this by presenting some examples of generalised parallelisable spaces and explicit constructions of suitable generalised frames on them in the case of O(d, d) generalised geometry [37,38].
By considering spaces that can be thought of as adjoint orbits, O(a), of some semi-simple Lie group G, with corresponding stabiliser groups H a , we show the normal bundles over these orbits, N O(a), in the case when O(a) is regular, are generalised parallelisable in the sense of [20]. An explicit O(d, d)-generalised frame is given for the generalised tangent bundle of this space, and a two-form B is constructed such that this frame satisfies the Leibniz conditions. In the case where the orbit is degenerate, which is equivalent to the stabiliser being non-abelian, construction of such a frame critically hinges on the existence of a suitable metric (see section 2.3). When the orbits are regular, the normal bundle is a flat bundle over the orbit and can be made compact, and they should thus admit a generalised Scherk-Schwartz reduction of type IIA, IIB and bosonic string theory [39] 3 .
Recently, an exposition of generalised Scherk-Schwartz uplifts of gauged supergravities was presented in [50], and generic classes of generalised Leibniz parallelisable spaces were presented. The spaces obtained here from regular adjoint orbits should fall under this classification.
The structure of this note is as follows: In the next section, the setup and conventions used herein will be presented, after which an action of G g on the normal bundle of the adjoint orbits will be constructed. In section 3, this will be used to explicitly construct a generalised Leibniz parallelisation of the O(d, d)-generalised tangent bundle of N O(a). In section 4, some explicit examples are presented, and section 5 concludes with a discussion.
2 Adjoint orbits and a G g action (Co-)adjoint orbits have been extensively studied in the literature (see for example chapter 8 of [51] or for a recent review, see [52]), and viewing the homogenous spaces G/H a as such orbits allows us to take advantage of known results about their properties, and especially their normal bundles. The adjoint orbits of a group G have at most dimension dim(G) − rank(G), and is then known as regular orbits. Orbits with smaller dimensions are said to be degenerate. Regular orbits are isoparametric submanifolds of G, which immediately gives that the normal bundles to these orbits are flat [53].

Conventions and preliminaries
Let G be a semi-simple Lie group, and H a subgroup of G, with corresponding Lie algebras g, h. We can as usual identify g with the tangent space at the origin of the group G. Furthermore, there is an Ad G -invariant scalar product on g, denoted by ·, · .
Let a be an element of the Lie algebra g. The orbit of a under the adjoint action of G is denoted O(a). Each orbit arising in this way may be parametrised by an element in the (closure of) the fundamental Weyl chamber of G, denoted by C C . Without loss of generality, we will always take a ∈ C. The regular orbits are given by a ∈ C, whereas for a ∈ C \ C, the orbit is degenerate. The different classes of orbits are therefore completely determined by the fundamental Weyl chamber of the group.
Consider the adjoint orbit through a point a ∈ C ⊂ g, and let H be the stabiliser subgroup of a. The orbit O(a) is then isomorphic to the homogenous space G/H, and the Lie algebra h can be identified with the annihilator of a, i.e. With respect to this inner product, there is an orthogonal decomposition of g as: which means that we can identify T Π(e) O(a) with v, where Π : G −→ G/H is the canonical projection.
We will here be interested in the normal bundle over this orbit, N O(a), which we view as N O(a) → g⊗g. That is, we parametrise the space by pairs (x, n) ∈ g⊗g, such that x parametrise the orbit and n the normal directions.
It is clear that we at any point x = Ad g a in this orbit will have: In the rest of this section, we will show that we can construct an action of G g on the normal bundle. In the next section, we will show that, under certain assumptions, this is enough for constructing a generalised frame on this space, allowing for a generalised Scherk-Schwartz reduction. These assumptions are trivially fulfilled in the case when O(a) is regular, in which case we present an explicit construction of the generalised frame. However, for degenerate orbits, the situation is more subtle and the existence of a generalised frame appears less clear.

An action of G g on M
Consider now a point in the normal bundle N O → g ⊗ g, given by: x ∈ g, n ∈ ann(x) h.
(We will from now on omit the a and only denote the orbit only by O for brevity.) In [54], the authors showed that there is a natural action of G g on the tangent space, T x O. This can be extended to the normal bundle as follows: let (g, η) ∈ G g, where the g should be viewed as a vector space, and the product be given by: The bracket in the Lie algebra of G g is given by: where (γ a ,γ a ) ∈ Lie (G g).
We then define the action on N x O by: where Π denotes the orthogonal projection with respect to the Ad G -invariant inner product ·, · .

The infinitesimal action
Let σ a ,σ a be a basis for the Lie algebra Lie (G g). The action of (2.5) give rise to 2 dim G vector fields, which in terms of the basis elements σ a ,σ a can be written as: Let the vector fields generated by the σ's be denoted byê v a and the ones generated by theσ's byê h a . In the case where ann(x) is abelian, (i.e. for a regular orbit O), these will be completely orthogonal in the sense of (2.2). In this case, we haveê v a ∈ ann(x) ⊥ v andê h a ∈ ann(x) h respectively, hence the notation. These satisfies the commutation relations of equation (2.4): However, even though their commutation relations are straight-forward, these vectors are not globally defined and cannot be frames for any viable metric. We can however, using their sums and differences, create two sets of dim G vectors that are globally defined, denoted byê ± a : It is easy to see that these indeed are globally defined, since [σ a , x] will only vanish when σ a ∈ ann(x), meaning that Π ann(x) σ a = σ a = 0. They satisfies the commutation relations:

The metric
We can define a metric using the vectorsê ± by requiering: For this to hold for both sets of vectors, the metric must satisfy Consider N O → g ⊗ g, and assume the metric is block diagonal in the two g-subspaces. If the orbit is regular, thenê h ,ê v lies in orthogonal subspaces of g in the sense of (2.2), and it is always possible to find a metric of the form in (2.10) on N O. If the orbit is degenerate, the situation is more subtle. If the metric still is block diagonal, the only situation where (2.11) is not trivially satisfied is when both σ a , σ b ∈ ann(x), in which case it takes the form: which vanishes if g restricted to ann(x) is Ad H -invariant, since n ∈ ann(x). It is however not a priori true that there is such an Ad H -invariant metric, satisfying (2.10). In the case when H is abelian, i.e. the orbit is regular, this can be shown to exist by straight-forward calculations.
If H is non-abelian, the situation is more subtle and the existence of such a metric needs to be verified.
It should be pointed out that the vectorsê ± in general are not killing vectors of the metric in (2.10). However, theê v are, which can be seen by a straight-forward calculation using the commutation relations of (2.7): The two sets of frames,ê ± are related by an orthogonal matrix P a b according toê + a = P a bê− b . This matrix encodes the inner product between the elements of the different frames, and is symmetric whenever O is regular.

Some basic O(d, d) generalised geometry
In order to proceed, we need to introduce some basic concepts of generalised geometry. On a manifold M , we denote the generalised tangent bundle by E ∼ T M ⊕ T * M . Elements of E are generalised vectors, V ,W , given by where v, w ∈ T M denote ordinary vectors and λ, µ ∈ T * M are one-forms. On this space, we define the generalised Lie derivative as: and we have a natural inner product (given by the O(d, d)-metric): where i v µ denotes contraction on the first index. In addition, there is also a generalised metric, G, which, in addition to the metric g depends on a two-form, B, and can be written in matrix notation as: This two-form B determines the map from T M ⊕ T * M to E via the exponential map: A convenient formula for our purposes will be the relation between the generalised Lie derivative and the exponential map: (2.15)

The generalised frame
We can define a generalised frame consisting of the ordinary framesê ± by: where iê± a g are the dual one-forms of the vectorsê ± a . It is straight-forward to show that these generalised frames are generalised orthonormal, i.e. they satisfy: However, for the purposes of consistent truncations, what we really need is to find a generalised Liebniz frame, so we want our generalised frame to satisfy the commutation relations of equation (2.9), that is: The vector parts of these relations are satisfied automatically, but not the one-form parts. We therefore define a three-form by assuming H = dB together with equation (2.15). That is, we define H by requiring the Liebniz relations (3.2) to hold. One can show that these all give equivalent definitions for H using the Jacobi identity, so it is enough for us to consider one, say for example LÊ+ aÊ + b : If this 3-form H is closed, this indeed gives us a two-form B via dB = H, and the frameÊ ± defines a generalised Liebniz prarallellisation of our space N O. Sinceê + is a globally defined frame, we can just as well define H through: To compute this is relatively straight forward, and results in: So the three-form H is given by Using the Jacobi identity, both in g and in Lie(G g), one can show that this is indeed a closed form, and so this defines a good two-form B via H = dB (and it indeed satisfies the other Liebniz relations as well). This calculation is spelled out in appendix A and hinges vitally on the orthogonality of the matrix P together with that the metric indeed can be defined uniquely by (2.10).

Some examples
It is well-known that the normal bundle to a regular adjoint orbit is globally flat [53]. Below, we will consider some examples, including both regular and degenerate orbits. Such orbits have been studied in for example [55] for some low-dimensional examples, and in [56] for G = SU (n), SO(n) and Sp(n).

G = SU (2)
In this case, there is only one type of adjoint orbit which, as a manifold, is isomorphic to S 2 . This is a regular adjoint orbit and its normal bundle hence trivial, allowing us to write the space N O SU (2) as a global product: This space was first shown to admit a consistent truncation of the bosonic string in [4], and an explicit generalised O(d, d)-frame was constructed on this space in [57], satisfying the Leibniz conditions as required. To make contact with the expressions presented therein, we can express the three-form dB in terms of the vectorsê h,v as: Here, h = u(1) is one-dimensional, so three-form becomes: Furthermore, since the vectorsê v a are given in terms of structure constants, (which is clear from the commutator in equation (2.8)), and since these in the SU (2) case are simply Levi-Civita's, the product of these in dH may be expanded in terms of delta-functions, after which it is clear that the term iêv a g ∧ iêv b g ∧ iêv c g-term above vanish, and the only surviving part of dB SU (2) is the mixed term betweenê h andê v , which indeed reproduces the expressions in [57] The R-factor can be compactified to an S 1 , allowing for a straight-forward compactification to be carried out on such an internal space, giving rise to a consistent truncation.

G = SU (3)
Here, there are two classes of orbits, the regular orbit O [55] , and the degenerate orbit O For the regular orbit, the stabiliser subgroup is abelian, and our construction of the generalised frame is straight-forward, arriving at a generalised Liebniz frame on: (4. 2) The normal bundle over the degenerate orbit, N O deg , will no longer be a global product, but rather consist of an SU (2) × U (1) principal bundle over a CP 2 base, giving us:

G = SO(4)
This is not a very interesting situation since it's double cover is given by SU (2) × SU (2). Therefore, the regular adjoint orbits of SO(4) will be diffeomorphic to S 2 × S 2 , so this will give us: The only non-trivial degenerate orbit will be diffeomorphic to S 2 , and the manifold N O SO(4) deg will then consist of SU (2) × U (1)-fibres over this.

G = SO(5)
Here, the regular orbit is given by where the space (2) is a real, 'flag-like manifold' [58]. There are a degenerate orbit corresponding to the real grassmannian RG 3,2 , whose normal space will be a principal SO(3) × SO(2)-bundle over RG 3,2 .
It could be argued that since dim(SO(5)) = 10, that this is not a very interesting example from the point of view of consistent truncations. However, O(d, d) generalised geometry captures the bosonic degrees of freedom, and should therefore also apply to bosonic string theory, making higher dimensions interesting.

Discussion and Outlook
The novel point of this note was to give an explicit construction of a generalised Liebniz frame of a new class of spaces given by normal bundles over adjoint orbits O of some group G. In the cases where the normal bundle is flat, this construction is complete and the normal directions can be compactified, and these manifolds should admit consistent truncations according to the recipe presented in [19]. In the case when this bundle is not trivial, the construction hinges on the existence of a suitable metric according to (2.10), and the compactification procedure required to obtain a low-energy effective theory is, at best, very subtle and is beyond the scope of this note.
Another line of attack closely related to the one presented herein is to think of the space O ∼ G/H as a submanifold of a local group manifold M ∼ G, and "zooming in" very close to such an orbit. If one zooms in enough, the manifold should look like N O. Since it is possible to construct a generalised frame on a local group manifold, it is reasonable to imagine this should also hold true on N O by such a construction, and the moduli one obtain in this way should be a limit of the moduli on the group manifold G. In the case when the stabiliser is abelian, it is indeed possible to carry out a similar construction to the one herein by considering Inönü-Wigner contractions [59] of the group G (and the stabiliser H), together with a rescaling of the metric and the normal coordinates. This procedure, however, appears to be dependent on a global splitting of the metric along, and orthogonal to, the orbit, which can only be done in the case where H is abelian.
In the recent work of [50], the scope of possible generalised Scherk-Schwarz reductions is elegantly considerd. This approach differs in that it is not primarily focused on consistent truncations but rather generalised Leibniz parallelisability on its own. The manifolds arising from regular orbits in this note should fall within the classification presented in [50], though the degenerate ones does not. It would be interesting to see if any connection can be made with the manifolds arising as normal bundles of degenerate orbits as well, and how the issues that arise in this construction is reflected in the setting of [50]. This is left to future work.