Super-Ehlers in Any Dimension

We classify the enhanced helicity symmetry of the Ehlers group to extended supergravity theories in any dimension. The vanishing character of the pseudo-Riemannian cosets occurring in this analysis is explained in terms of Poincar\'e duality. The latter resides in the nature of regularly embedded quotient subgroups which are non-compact rank preserving.


Introduction
Three decades ago it was shown [1] that the D-dimensional Ehlers group SL(D − 2, R) is a symmetry of D-dimensional Einstein gravity, provided that the theory is formulated in the light-cone gauge. For any D 4-dimensional Lorentzian space-time, this results enables to identify the graviton degrees of freedom with the Riemannian coset even if the action of the theory is not simply the sigma model action on this coset (with the exception of a theory reduced to D = 3). In D = 4, this statement reduces to the well known fact that the massless graviton described by the Einstein-Hilbert action with two degrees of freedom of ±2 helicity has an enhanced symmetry SO(2) → SL (2, R).
In N -extended supergravity in D dimensions, U -duality 1 symmetries play an important role to uncover, in terms of geometrical constructions, the non-linear structure of the theories, whose most symmetric one is the theory with maximal supersymmetry (2N = 32 supersymmetries). Furthermore, U -duality symmetries get unified with the Ehlers space-time symmetry if one descends to D = 3 [4,5]. In the maximal case, the D = 3 U -duality group is E 8 (8) , with maximal compact subgroup (mcs) SO (16), which is also the underlying Clifford algebra for massless supermultiplets of maximal supersymmetry. As a consequence, the bosonic sector of the theory is described by the sigma model E 8 (8) /SO (16) [7,8,6].
Following these preliminaries, it comes as no surprise that it was further discovered that in lightcone Hamiltonian formulation maximal supergravity exhibits E 7 (7) symmetry in D = 4 [10] and E 8(8) symmetry in D = 3 [11] (for the D = 11 theory, see [9]). Indeed, in any space-time dimension D and for any number of supersymmetries N = 2N , it is known that the D = 3 U -duality group G 3 N [12] embeds (through a rank-preserving embedding; for some basic definitions, see the start of App. A) the Ehlers group SL(D − 2, R) as a commutant of the U -duality group G D N [15,16]: It is then natural to conjecture that in a suitable light-cone formulation of any N -extended supergravity theories G D N × SL(D − 2, R) (which we dub super-Ehlers group) is a manifest symmetry of the theory. Even if the super-Ehlers group is a bosonic extension of the Ehlers group itself, the presence of the U -duality commutant G D N in (1.2) is closely related to supersymmetry. It is intriguing to notice that the super-Ehlers symmetries, which we classify below in any dimension, sometimes exhibit an "enhancement" into some larger group 2 ; this occurs whenever the embedding (1.2) is non-maximal, and in D = 10 type IIB supergravity. Furthermore, it sometimes occurs that the embedding (1.2) is maximal but non-symmetric, as in D = 11 supergravity.
At any rate, we will show that the common features of the embedding (1.2) are at least two (cfr. the start of App. A): • It is regular and preserves the rank of the group. Indeed, it generally holds that The same relation holds for the non-compact rank of these groups, namely the rank of the corresponding symmetric Riemannian manifolds of which the groups encode the isometries:

4)
1 Here U -duality is referred to as the "continuous" symmetries of [2]. Their discrete versions are the U -duality nonperturbative string theory symmetries introduced by Hull and Townsend [3]. 2 For enhancement to infinite symmetries, see [17].
where H 3 N and H D N are the maximal compact subgroups of G 3 N and G D N , respectively. As mentioned above, this does not imply the embeddings to be in general maximal nor symmetric.
• The pseudo-Riemannian coset resulting from (1.2) has always zero character [13,14], namely it has the same number of compact and non-compact generators. We will show that this latter property is related to the Poincaré duality (alias electric-magnetic duality) of the spectrum of massless p > 0 forms of the theory, which can essentially be traced back to the existence of an Hodge involution in the cohomology of the scalar manifold, singling out only the physical forms and their duals in the cohomology of the (D − 2)-dimensional transverse space. This property also follows from the regularity of the embedding of G D N × SL(D − 2) inside G 3 N , the semisimplicity of the two groups and properties (1.3), (1.4), as it will be shown in Appendix A.3.
There is also another aspect of interest in the present analysis : the role played by exceptional Lie groups and their relation to Jordan algebras and Freudenthal triple systems [18,19]. In particular, a mathematical construction, called Jordan pairs (see e.g. [20] for a recent treatment, and a list of Refs.) corresponds to the maximal non-symmetric embedding E 8(8) ⊃ E 6(6) × SL(3, R), (1.5) which is nothing but (1.2) specified for maximal supersymmetry (N = 16) and D = 5. We point out that the Jordan pairs relevant for supergravity theories always pertain to suitable non-compact real forms of Lie algebras, differently e.g. from the treatment given in [20]. Moreover, it is worth observing that in D = 11 supergravity G 11 16 is empty, and thus (1.2) is the following maximal non-symmetric embedding [4]: (1.6) which in fact was used long time ago [21] in order to construct the gravity multiplet of this theory [22]. For maximal supergravity (N = N max = 16), (1.2) reads 3 where G D 16 = E 11−D(11−D) denotes the so-called Cremmer-Julia sequence [7,8]. The unique exception is provided by type IIB chiral D = 10 supergravity, in which (1.2) is given by a two-step chain of maximal embeddings 4 : E 8(8) ⊃ s SL(2, R) × E 7(7) ⊃ s SL(2, R) × SL(8, R), (1.8) which preserves the group rank.
The plan of the paper is as follows.
In Sec. 2 we start by recalling some basic facts on SO(N ) Clifford algebras relevant for the classification of massless multiplets of N -extended supersymmetry in any dimension. Here N = 2N denotes the number of supersymmetries, regardless of the dimension D. Thus, for instance maximal supergravity corresponds to N = 32 (8 spinor supercharges in D = 4), whereas the minimal supergravity we consider has N = 8 (2 spinor supercharges in D = 4). We then proceed to considering the embedding (1.7) pertaining to maximal supergravity in any dimension D 4 (in D = 10 both IIA and IIB theories are considered). The embedding (1.2), which can be regarded as the "non-compact enhancement" of Nahm's analysis [21], in all cases consistently provides the massless spectrum of the corresponding theory with the correct spin-statistics content; illustrative analysis is worked out for D = 11 and D = 10 maximal theories. Other theories which do not exhibit matter coupling are also considered, namely N = 10, 12 in D = 4 and N = 12 in D = 5.
In Sec. 3 we consider half-maximal supergravity theories (N = 8), which can be matter coupled and exist in all D 10 dimensions; for D = 6 we consider both inequivalent theories, namely the chiral (2, 0) (type IIB) and the non-chiral (1, 1) (type IIA) ones. Theories with N = 6, living in D = 4, are also considered.
Then, in Sec. 4 we consider quarter-maximal theories (N = 4), which live in D = 4, 5, 6 and admit two different kinds of matter multiplets. We confine ourselves to theories with symmetric scalar manifolds, which (apart from the minimally coupled models in D = 4 and the non-Jordan symmetric sequence in D = 5) admit an interpretation in terms of Euclidean Jordan algebras.
Pseudo-Riemannian cosets associated to the maximal-rank embeddings (1.2) are then analyzed in Sec. 5. All such cosets enjoy the property of having the same number of compact and non-compact generators. This is also proven, using general group theoretical arguments, in Appendix A.3. In Subsec. 5.2 this property is related to the invariance of the spectrum of massless bosonic p > 0 forms under Poincaré-duality, or equivalently in Subsec. 5.3 in terms of an Hodge involution acting on the coset cohomology.
Final remarks and outlook are given in Sec. 6. Three Appendices conclude the paper. In App. A, some embeddings of non-compact, real forms relevant for our analysis are rigorously proved, while in App. B the issue of inequivalent "dual pairs" of subalgebras of the U -duality algebra is discussed (see also [23]). The related notions of T -dualities as so (8,8) outer-automorphisms are also dealt with. In App. C the issue of Poincaré duality is revisited with an explicit algebraic construction which makes use of appropriate level decompositions.

Clifford Algebras and "Pure" Theories
In the seminal paper by Nahm [21], it was shown how massless multiplets of supergravity are built in terms of irreps. of SO(D−2), the little group (spin) of massless particles in D dimensions. The number of supersymmetries 2N is encoded in the Clifford algebra of SO(N ), and therefore the supermultiplets can be regarded as SO (N ) spinors decomposed into SO (D − 2) irreps (for theories with particles with spin s 2, which we consider throughout, N max = 16). Bosons and fermions thus correspond to the two semi-spinors (or chiral spinors) of 5 SO(N ).
In any dimension D 4, SO (N ) exhibits a certain commuting factor with the massless little group SO (D − 2). For "pure" supergravities, in which only the gravity multiplet is present, such a commuting factor is the so-called R-symmetry of the theory. Then the question arises as to which is the non-compact group commuting with the SL(D − 2, R) Ehlers group (which thus extends the massless little group including the R-symmetry), and furthermore which is the non-compact group which extends the SO(N ) of the N -dimensional Clifford algebra pertaining to 2N local supersymmetries.
In describing massless multiplets of theories with N = 2N local supersymmetries, one consider the the rest-frame supersymmetry algebra without central extension. Since the momentum squares to zero (P µ P µ = 0), only half of the supersymmetry charges survive, and the creation operators of N charges describe an SO(N ) Clifford algebra. Moreover, due to the fact that in D 4 spinors always have real dimension multiple of 4, N is always even : N = 4, 6, 8, 10, 12, 16 (we do not consider here N = 2 at D = 4, namely minimal 4-dimensional supergravity with 1 spinor supercharge). It thus comes as no surprise that U -duality groups G 3 N in D = 3 (in which there is only distinction between bosons and fermions, but no spin is present for massless states) contain in their mcs the Clifford algebra symmetry SO(N ).
Supersymmetry dictates that massless bosons and fermions are simply the two (chiral, semi-) 5 Note that N is always even, since for D 4 spinor charges have real dimensions multiples of 4. Furthermore, it should be remarked that the cases D = 4, N = 2 and D = 10, N = 8 are somewhat particular, because N = D − 2, so the two Clifford spinors directly provide bosonic and fermionic supermultiplets' representations. spinor irreps. of SO(N ), while their spin s content in D space-time dimensions is obtained by suitably branching such irreps. into SO(D − 2), which is the little group (spin) for massless particles in D dimensions.
In the present Section we consider "pure" theories in which the matter coupling is not allowed; they include maximally supersymmetric (N = 16) theories in any dimension D 11, as well as nonmaximal theories with N = 10, 12 in D = 4 and N = 12 in D = 5. For such theories, the Clifford algebra SO(N ) is nothing but the mcs of the U -duality group G 3 N in D = 3; for non-maximal theories (N < 16), this is true up to the presence of the so-called Clifford vacuum factor group, which expresses further degeneracy of the Clifford algebra symmetry. Moreover, the group H D N = mcs G D N which commutes with SO(D − 2) inside SO(N ) is the R-symmetry, providing the degeneracy of the spin s representations in the decomposition of the chiral spinors under the embedding 6 [21] which is the (not necessarily maximal-rank, nor maximal nor symmetric) counterpart of (1.2) at the level of mcs. The subscript "J" denotes the spin group throughout.

N = 16 (Maximal )
For maximal (N = 16) supergravity theories with massless particles, the D = 3 U -duality group is G 16 3 = E 8 (8) , with mcs SO (16), which is the Clifford algebra for massless particles with N = 32 supersymmetries. (1.7) provides the rank-preserving embedding of D-dimensional Ehlers group , belonging to the so-called the Cremmer-Julia sequence. All cases in 4 D 11 dimensions are reported in Table 1 (non-compact level (1.2)-(1.7)) and in Table 2 (mcs level (2.1)). In particular, in Table 2 also the decomposition of the vector irrep. 16 of the Clifford algebra SO(16) = mcs E 8 (8) of maximal (N = 16 → N = 32) supersymmetry is reported for the embedding (2.1) pertaining to this case, namely [21] (see also [24]): where, as mentioned above, R 16 D ≡ mcs G 16 D ≡ H 16 D is the R-symmetry of the maximal supergravity in D (Lorentzian) space-time dimensions. Note that the irrep. of SO(D − 2) occurring in the branching of the 16 along (2.2) are all spinors, and the R-symmetry R 16 D is real, pseudo-real (quaternionic), complex, depending on whether such spinor irrep. is real, pseudo-real or complex, respectively.
Let us scan them briefly (as anticipated, for D = 11 and D = 10 the massless spectrum analysis is also worked out, as an example of the consistence of the embeddings with the massless spectrum of the corresponding theory). For convenience of the reader, we anticipate that the embeddings (1.2) and (2.1) are maximal in D = 11, 7, 5 (non-symmetric) and 4 (symmetric), while they are next-to-maximal in D = 10,9,8,6; in these latter cases, an "enhancement" of E 11−D(11−D) × SL (D − 2, R) occurs (see analysis below).
1. D = 11 (M -theory). There is no continuous U -duality (and thus R-symmetry) group, and (1.7) specifies to (1.6), namely the maximal non-symmetric embedding of the Ehlers group SL(9, R) only: where 84 and 84 ′ are the 3-fold antisymmetric of SL(9, R) and its dual; they correspond to gauge fields coupling to M 2 branes and M 5 branes, respectively. The corresponding mcs level 6 Further commuting factor group occurs in the l.h.s. of (2.1) in non-maximal (N 16) theories; see analysis below.
2) for maximal supergravity theories (N = 16) in 11 D 4 Lorentzian space-time dimensions [15,16]. G D N is the U -duality group in D dimensions for the theory with N = 2N supersymmetries. SL(D − 2, R) is the Ehlers group in D dimensions. For N = 16, G 3 16 = E 8 (8) , and G D N = E 11−D (11−D) belongs to the Cremmer-Julia sequence; thus, (1.7) is obtained. The type (max(imal), n(ext-to-)m(aximal), s(ymmetric), n(on-)s(ymmetric)) of embedding is indicated. Explicit proofs are given in App. A. is given by the specification of (2.1) to the following non-symmetric embedding of the massless spin group SO (9) only: SO (  note the "enhancement" to SL(9, R), consistent with the M -theoretical origin of IIA theory. The corresponding mcs level is where SO(8) is the massless spin group. Throughout our analysis, we dub "next-to-maximal" (nm) those embeddings given by a chain of two maximal embeddings; note that all nm embeddings considered in the present investigation are of maximal rank, namely they preserve the rank of the original group. For what concerns the IIA massless spectrum, one considers the branchings of 128 (bosons) and 128 ′ (fermions) of the Clifford algebra SO (16) under the nm embedding (2.8): where the subscripts "v", "s" and "c" respectively stand for vector, spinor, conjugate spinor, and they pertain to the triality of SO(8), the little group (spin group) of massless particles in D = 10. 56 i , 28, 35 i and 8 i (i = v, s, c) are the rank-3 antisymmetric, adjoint, rank-2 symmetric traceless and vector/spinor irreps. of SO(8), respectively. Thus, the branching (2.9) consistently pertains to the IIA massless bosonic spectrum : 3-form C  3. On the other hand, in D = 10 type IIB theory the U -duality is G 10 IIB
6. In D = 7 the U -duality is G 7 16 = SL(5, R) ≡ E 4 (4) , and its mcs is the R-symmetry SO(5) ∼ U Sp (4). The relevant embedding is maximal non-symmetric: Note that in this case there is perfect symmetry between the R-symmetry and the massless spin sectors. 8 (2.21) is the n = 4 case of the maximal non-symmetric embedding pattern where A2,0 is the rank-2 antisymmetric skew-traceless irrep. of U Sp (2n). For the first appearance of such an embedding in supersymmetry, see [25].
8. In D = 5 the U -duality undergoes an exceptional enhancement : G 5 16 = E 6(6) , and its mcs is the R-symmetry U Sp (8). The relevant embedding is maximal non-symmetric, and it is given by (1.5), which we report here (note that it is the first step of nm embedding (2.20)-(2.21)): The corresponding Jordan algebra interpretation of (2.27) is as follows: and it is a particular non-compact, real version of the Jordan-pair embeddings of exceptional Lie algebras recently considered in [20]. Note that the SU (2) in (2.28) is the principal SU (2) in SL(3, R) in (2.27). 9. In D = 4 the U -duality is G 4 16 = E 7 (7) , and its mcs is the R-symmetry SU (8). The relevant embedding is maximal symmetric (note that it is the first step of chains (2.11)-(2.12) and (2.18)-(2.19)): The corresponding Jordan algebra interpretation of (2.30) is as follows: where Conf denotes the conformal group of J Os 3 (see e.g. [19] and Refs. therein). Similar Jordan-algebraic interpretations can be given for other supergravities in various dimensions. 9 Subscripts "L" and "R" denote left and right chirality, respectively. 10 In theories related to Euclidean Jordan algebras J3 of rank 3, the quasi-conformal QConf (J3), conformal Conf (J3) and reduced structure Str0 (J3) groups are the U -duality groups in D = 3, 4 and 5 dimensions, respectively. In particular, Conf (J3) is nothing but the automorphism group Aut (M (J3)) of the corresponding Freudenthal triple system [18,19].

N = 12
In the "pure" theory with N = 12, the D = 3 U -duality group is G 3 12 = E 7(−5) , with mcs SO(12) × SU (2) CV , where SO(12) is the Clifford algebra for massless particles with N = 24 supersymmetries. The SU (2) CV factor pertains to the so-called Clifford vacuum (CV ), which is generally present for non-maximal theories (N < 16), and it indicates further degeneracy of the Clifford algebra symmetry. In this case, SU (2) CV can be also explained by recalling that this theory shares the very same bosonic sector of a matter-coupled supergravity with N = 4 [18], in which it is the R-symmetry of the hypermultiplets' sector.
This theory can consistently be uplifted only to D = 4 and D = 5.
1. In D = 5 the U -duality is G 5 12 = SU * (6), and its mcs is the R-symmetry U Sp (6). The relevant embedding is maximal non-symmetric: (2.33) where we introduced the subscript "J" in order to discriminate between the Clifford vacuum SU (2) CV and the SU (2) J pertaining to the massless spin group in D = 5. Note that the embedding (2.33) is maximal non-symmetric, while the embedding (2.34) is non-maximal nonsymmetric.
2. In D = 4 the U -duality is G 4 12 = SO * (12), and its mcs is the R-symmetry U (6). The relevant embedding is maximal symmetric: and it pertains to the so-called c * -map (see e.g. [27], and Refs. therein).
where m is the number of matter multiplets in D = 3 other than those coming from the reduction of the gravity multiplet in D dimensions. Furthermore, mcs G 3 (8) is the Clifford algebra for massless particles with N = 16 supersymmetries, and SO (D − 2 + m) CV is the Clifford vacuum symmetry.
The relevant chain of maximal embeddings leading to the embedding of the D-dimensional Ehlers group SL (D − 2, R) into SO (8, D − 2 + m) depends on the dimension and on the type of theory. We anticipate that embeddings (1.2) and (2.1) are maximal in D = 4 (symmetric) and next-to-maximal in 5 D 10.
• For D 5 (and D = 6 type IIA (1, 1)), it is given by the following chain of two maximal symmetric steps: . Furthermore, it is worth remarking that also for m = 0 the Clifford vacuum degeneracy is still present with an SO (D − 2) CV factor; this is an extra spin quantum number carried by the SO(8) Clifford algebra spinor. In fact, by considering the mcs level of the chain (3.1), one obtains . SO (m) CV is the part of Clifford vacuum symmetry due to matter coupling.
• For D = 4, the maximal symmetric embedding reads: and it pertains to the so-called c * -map (see e.g. [27], and Refs. therein). The group commuting with SL(2, R) Ehlers inside SO (8, 2 + m) is the 4-dimensional U -duality group G 4 8 = SL(2, R) × SO (6, m). Also in this case for m = 0 the Clifford vacuum degeneracy is still present with an SO (2) CV factor. In fact, by considering the mcs level of (3.3), one obtains the following maximal symmetric embedding (SO(6) ∼ SU (4)): Moreover, SO (m) CV is the part of Clifford vacuum symmetry due to matter coupling.
• For D = 6 type IIB (2, 0), it suffices to start with SO (8, 3 + m), and the maximal symmetric embedding reads as follows: The group commuting with SL (4, . The corresponding mcs level reads where SO(4) J = mcs (SL(4, R) Ehlers ), and the R-symmetry is SO(5) ∼ U Sp(4). Furthermore, SO (m) CV is the part of Clifford vacuum symmetry due to matter coupling.
All cases in 4 D 10 dimensions are reported in Tables 3 and 4.
The D = 3 U -duality group is G 3 6 = SU (4, m + 1), where m is the number of matter multiplets in D = 3 other than those coming from the reduction of the gravity multiplet in 4 dimensions. Furthermore, mcs G 3 (6) is the Clifford algebra for massless particles with N = 12 supersymmetries, and U (m + 1) CV is the Clifford vacuum symmetry.
The embedding of the 4-dimensional Ehlers group SL (2, R) into SU (4, m + 1) is maximal and symmetric: and at the mcs level: Clifford vacuum symmetry, which is related to the number of matter multiplets.

N = Matter Coupled Symmetric Theories
Quarter-maximal theories with N = 4 exist in 3 D 6; in particular, in D = 6 they are chiral (1, 0) theories. The new feature of N = 4 theories is the possible existence of two different types of matter multiplets, namely vector and hyper multiplets, transforming in different ways under the R-symmetry, which is U (2) in D = 4 and U Sp(2) in D = 5, 6. In the following treatment, we will only consider theories based on symmetric Abelian-vector multiplets' scalar manifolds, which is a restriction to D = 4 (Kähler) and D = 5 (real) special geometry; these theories will be denoted as 11 symmetric N = 4 theories.
In D = 4, 5, symmetric theories are classified by two infinite sequences, as well as by isolated cases given by the so-called "magical" models.
We will also not consider the (D-independent) hypermultiplets' quaternionic scalar manifolds. 11 We will not consider here the so-called non-Jordan symmetric sequence (see e.g. [28] and Refs. therein) in D = 5, based on vector multiplets' real special symmetric scalar manifolds SO(1,n) SO(n) , which gives rise to non-symmetric coset manifolds in D = 4 and in D = 3.
For N = 4, we recall that the Clifford algebra decomposes as where SU (2) v pertains to the D = 3 reduction of D = 4 vector multiplets, while SU (2) h is related to the hypermultiplet sector, which is insensitive to the number of space-time dimensions in which the quarter-maximal theory is defined (namely, 3 D 6). Since we disregard hypermultiplets, in the treatment below we only consider SU (2) v (and thus we remove the subscript "v"), which will be a commuting factor in the mcs of the D = 3 U -duality group G 4 4 .

Minimal Coupling Infinite Sequence and "Pure" D = 4 Supergravity
We start by considering the infinite sequence of D = 3 quaternionic Kähler symmetric spaces which can be uplifted only to D = 4, giving rise to Maxwell-Einstein supergravity models minimally coupled to n vector multiplets [31]. The D = 3 U -duality group is G 3 4 = SU (2, 1 + n). The embedding of the 4-dimensional Ehlers group SL (2, R) into SU (2, n + 1) is maximal and symmetric: and at the mcs level: under which the D = 4 vector multiplets are charged, whereas the U (n) factor correspond to D = 4 Clifford vacuum symmetry (completely due to matter coupling).
By merging (4.3) and (4.4), the following c-map is obtained [29]: where CP n denotes the complex projective (non-compact) spaces. Note that for n = 0 the quaternionic manifold (4.2) is not only Kähler, but also special Kähler, and it is an example of Einstein space with self-dual Weyl curvature (see e.g. [32], and Refs. therein). It is usually called the universal hypermultiplet, and it corresponds to the c-map of "pure" N = 2 supergravity in D = 4, obtained as "n = 0 limit" of the CP n sequence; namely, by specifying n = 0 in (4.5) [29]: .
The corresponding Jordan algebra interpretation of (4.13) reads because the ST 2 model is related to the (non-generic) semi-simple rank-3 Euclidean Jordan algebra given by R ⊕ Γ 1,0 ∼ R ⊕ R.

The Jordan Symmetric Infinite Sequence
The aforementioned ST 2 model is actually the first element of the so-called Jordan symmetric sequence of quarter-maximal theories.
where n is the number of matter multiplets in D = 3 other than those coming from the reduction of the gravity multiplet in D dimensions. Furthermore, mcs G 3 (2) h is the Clifford algebra for massless particles with N = 8 supersymmetries, and SO (D − 2 + n) CV is the Clifford vacuum symmetry.
Let us consider the relevant chain of maximal embeddings leading to the embedding of the Ddimensional Ehlers group 14

D = 6
In D = 6, it suffices to start from G 3 4 = SO (4, 3 + n), and the corresponding maximal symmetric embedding reads (4.17) and at the mcs level: where n is the number of matter (tensor) multiplets in D = 6. The group commuting with SL (4, R) inside SO (4, 3 + n) is nothing but the 6-dimensional U -duality group of tensor multiplets G 6 4 = SO(1, n).

D = 5
For D = 5, one branches once more from (4.17), getting: and at the mcs level: where n+1 is the number of matter (vector) multiplets in D = 5. The group commuting with

Magical Models
Let us now consider the isolated cases of symmetric N = 8 quarter-maximal theories, the so-called magical models [18]. They are associated to rank-2 (in D = 6) and rank-3 (in D = 5) Euclidean Jordan algebras over the four normed division algebras O (octonions), H (quaternions), C (complex numbers) and R (real numbers), and to the Freudenthal triple systems over such algebras (in D = 4). Consequently, they can be parametrized in terms of the real dimension of the relevant division algebra, namely q = 8, 4, 2, 1 for O, H, C and R, respectively. In this respect, the T 3 model treated above corresponds to q = −2/3. We will now analyze the relevant embeddings in D = 4, 5 and 6.

D = 4
In D = 4, the magic models are related to the Freudenthal triple system M J A 3 over the rank-3 simple Euclidean Jordan algebra J A 3 (A = O, H, C, R). The D = 3 and D = 4 U -duality groups are nothing but the quasi-conformal and conformal group of J A 3 , respectively, and they are related by the following maximal symmetric embedding: with mcs level involving the D = 4 massless spin group: . (4.26) The various cases are listed in Tables 5 and 6.

D = 5
In D = 5, the magic models are related to J A 3 's themselves. The D = 5 U -duality group is the reduced structure group of J A 3 , and the embedding of the D = 5 Ehlers group SL(3, R) into the D = 3 U -duality group is maximal and non-symmetric: with mcs level involving the D = 5 massless spin group: The various cases are listed in Tables 7 and 8.

D = 6
In D = 6, the magic models are related to the rank-2 Jordan algebra J A 2 ∼ Γ 1,q+1 (where "∼" here denotes a vector space isomorphism). Namely, the D = 6 U -duality group is nothing but the reduced structure group of J A 2 itself, with the exception of the cases corresponding to q = 4 and q = 2, which have a further factor 15 A q=2 = SO(3) resp. A q=1 = SO(2) in the U -duality group. The embedding of the D = 6 Ehlers group SL(4, R) into the D = 3 U -duality group is obtained by a two-steps chain of maximal and symmetric embeddings (A q = Id, SO(3), SO(2), Id respectively for q = 8, 4, 2, 1): with mcs level involving the D = 6 massless spin group: Note the "enhancement" to SO (4, q + 4) × A q in (4.29). The various cases are listed in Tables 9  and 10.  (2), whereas the subscript "D" stands for diagonal embedding

Cosets with χ = 0 and Poincaré Duality
From the previous treatment, a class of non-compact, pseudo-Riemannian homogeneous spaces can be naturally constructed, with general structure: determined by the embedding of the direct product of the D-dimensional Ehlers group SL(D − 2, R) and of the D-dimensional U -duality group G D N of a supergravity with N = 2N supersymmetries into the U -duality group of the same theory reduced to D = 3 (Lorentzian) space-time dimensions. From previous Secs., such an embedding can be maximal or non-maximal (namely, next-to-maximal), and symmetric or non-symmetric, but, as mentioned, it always preserves the rank of the group (1.3), as well as the non-compact rank of the D = 3 coset G 3 N /H 3 N (1.4). Interestingly, the cosets M D N 's (5.1) all share the same feature : they have an equal number of compact and non-compact generators, thus implying the their coset character χ [14,13] to be vanishing: This property can also be related to the "mcs counterpart" of the class of cosets (5.1), given by the compact, Riemannian homogeneous spaces with general structure This is a consequence of (5.2) as well as the general formula on the signature of a pseudo-Riemannian coset G/H (see e.g. [14]) Along this line, further elaboration is possible. Indeed, it generally holds that .  Table 11, along with their number of compact and non-compact generators. Among M D 16 's, the unique maximal and symmetric coset is the one pertaining to D = 4 (cfr. (2.30)): which is a rank-4 para-quaternionic space, as resulting from the classification of [35]. Also the corresponding   The values of c = nc given in (5.11) and (5.13) match the ones of the magical quarter-maximal (N = 4) theory for q = 4 (see (5.40) and (5.44), respectively); indeed, these theories share the same bosonic sector, and they are both related to J H 3 .

N = 10
The specification of (5.1) and (5.3) to supergravity with N = 10 in D = 4 gives rise to the following symmetric spaces The specification of (5.1) and (5.3) to half-maximal supergravity (N = 8) gives rise to the following spaces ; (5.18) they are listed in Table 12, along with their number of compact and non-compact generators. Among M D 8 's, the unique maximal and symmetric cosets are the ones pertaining to D = 6 IIB and D = 4(cfr. (3.3)):  The specification of (5.1) and (5.3) to supergravity with N = 6 in D = 4 gives rise to the following symmetric spaces , c = nc = 2m + 7; (5.21) , c = nc = 2n + 2; (5.23) M 4 4 has rank 1 for n = 0, and rank 2 for n 1, and it is para-quaternionic. It is nothing but a suitable pseudo-Riemannian form of the manifold (4.2) itself, namely the c * -map of the rank-1 symmetric special Kähler maximal coset in D = 4: .
It is nothing but a suitable pseudo-Riemannian form of the manifold in the r.h.s. of (4.11), namely the c * -map of the rank-1 symmetric special Kähler maximal coset in D = 4: .

Poincaré Duality
We are now going to analyze the signature split of the manifolds M D N (5.1), focussing on the maximal (N = 32) and magical quarter-maximal cases (N = 8).

Hodge Involution and Coset Cohomology
, (5.103) where g N 3 and g N D respectively are the Lie algebras of G N 3 and G N D , and n d is the (real) dimension of the relevant irreps. of the U -duality group G N D in D dimensions. Note that the r.h.s. of (5.103) is manifestly invariant under the Hodge involution * (5.102). Thus, the vanishing character (5.2) of cosets M D N 's (5.1) trivially follows from (5.108) (II) In maximal D = 10 IIA supergravity, the relevant values are d = 1, 2, 3 with n 1 = n 2 = n 3 = 1, and thus (5.103) and (5.104) specifies as follows: In terms of the Cartan decomposition of the non-maximal non-symmetric Riemannian compact coset M 10 IIA

16
= SO(16)/SO(8), the result (5.110) can be obtained as a consequence of the next-to-maximal non-symmetric embedding (Adj = Λ 2 ; (cfr. (2.2) and Table 2) so(16) ⊃ ns so(8) Finally, we present below the same analysis for other two "pure" supergravities:  (6) , (5.117) where the D = 5 massless spin algebra su (2) is not modded out in order to determine M 5 12 , and it corresponds to the "extra" U Sp(6) (R-symmetry-)singlet, a peculiar feature of this extended supergravity theory (which makes it amenable to an N = 4 interpretation). where the subscripts denote the charges with respect to the D = 4 massless spin algebra u(1).

A.2 Other Embeddings
Embeddings considered here were also dealt with in [16]. Here we provide a detailed and explicit construction of a number of embeddings in terms of the generators of the corresponding Lie algebras, using the notation of [13]. Let us start discussing in detail the embeddings of E 6(−26) × SL(3, R) and SO (1,9) × SL(4, R) inside E 8(−24) . At the level of the corresponding Lie algebras, these embeddings are illustrated in Figure 3, where the Satake diagrams of e 6(−26) ⊕ sl(3, R) and so (1,9) ⊕ sl(4, R) are obtained from the e 8(−24) one once again using Dynkin's procedure of extending the latter and canceling a suitable simple root. Let us briefly review the definition of Satake diagrams for nonsplit (i.e. non-maximally-non-compact) Lie algebras and the construction of the e 6(−26) ⊕ sl(3, R) and so (1,9) 20 We shall omit the prime in the following. 21 We can always find a suitable basis for the matrix representation of the generators so that τ (M ) = −M † . This means that we shall regard compactness and non-compactness of a generator to be synonyms, in any matrix representation, of being anti-hermitian and hermitian, respectively. Moreover, in our conventions, E−α = −τ (Eα) = E † α . Note that h c is the Cartan subalgebra of an so(8) subalgebra of e 8(−24) whose Dynkin diagram is defined by the black roots in Fig. 3. The e 8 positive roots split into a 12-dimensional sub-space ∆ The conjugation σ with respect to e 8(−24) is the conjugation on the complex e 8 which leaves the elements of the subalgebra e 8(−24) invariant. It defines a correspondence between e 8 -roots α ↔ α σ such that σ(E α ) ∝ E α σ . The couple of roots α , α σ satisfies the property: Clearly if α ∈ ∆ which can be both brought to an upper-triangular form, for all α. If, on the other hand, α ∈ ∆ (0) To summarize, the e 8(−24) generators can be expressed in terms of the e 8 canonical basis as follows: (A.16) The 112-dimensional solvable Lie algebra s 0 = h nc ⊕l + is the one defined by the Iwasawa decomposition of e 8(−24) with respect to e 7(−133) ⊕ su (2), and its generators, in a suitable basis, can all be represented by upper-triangular matrices. The centralizer of h nc is the so(8) subalgebra given by h c ⊕ m 0 and is also contained inside the subalgebras e 6(−26) and so (1,9), as it is apparent from Fig. 3. The e 6(−26) generators in terms of the above e 8(−24) ones are easily written: (A. 17) where ∆ + [e 6 ] are the e 6 -positive roots in the e 8 -root system, while The sl(3, R) subalgebra commuting with e 6(−26) has the following form: note that β σ x = β x . Finally the so(1, 9) ⊂ e 6(−26) generators read: (A. 19) where ∆ + [so (10)] are the roots of the complex so(10) algebra within e 8 -root system, and The sl(4, R) subalgebra commuting with so(1, 9) is described by the following generators: By the same token we can prove other embeddings, like SL(3, R) × SU * (6) ⊂ E 7(−5) and SL(3, C) × SL(3, R) ⊂ E 6(2) , see Fig. 4. In the latter case there is a subtlety which is not apparent from the truncation of the extended Satake diagram: The bottom-right diagram in Fig. 4 would naively suggest that the roots α 1 , α 3 , α 5 , α 6 define two commuting sl(3, R) subalgebras. This is however not the case since, as represented by the lower arrows, the conjugation σ corresponding to the real form e 6(2) inside the complex e 6 , maps α 1 and α 3 into α σ 1 = α 6 and α σ 3 = α 5 , respectively. As a consequence of this the e 6 shift generators corresponding to the two orthogonal sl(3, R) root spaces are mixed together in σ-invariant combinations inside e 6(2) , which make the shift generators of a sl(3, C) subalgebra. This subalgebra also contains the two non-compact combinations H α 1 + H α 6 , H α 2 + H α 5 and the two compact combinations i(H α 1 − H α 6 ), i (H α 2 − H α 5 ) of the e 6 Cartan generators.

A.3 General Features
One can generalize the above discussion and show that, as a general feature of the embeddings considered in this work, the g 3 N algebra, and its super-Ehlers subalgebra g D N ⊕ sl(D − 2) can be written in the forms: can be chosen to coincide, where dim(h nc ) is the non-compact rank of the two groups. This is implicit in Dynkin's construction of the g D N ⊕ sl(D − 2) algebra by truncating the extended diagram of g 3 N . Moreover the centralizer of h nc , which is the compact algebra h c ⊕ m 0 , is common to the two algebras: For a split (maximally non-compact) g 3 N , h c = m 0 = ∅ and α σ = α. The nilpotent spaces l ± ,l ± have the form: where, as usual,∆ + [g 3 N ] denotes the set of positive roots of the (complexification of) g 3 N with nontrivial restriction to h nc , and ∆ + [g D N ⊕ sl(D − 2)] the set of positive roots of the (complexification of) g D N ⊕ sl(D − 2), which is a subset of ∆ + [g 3 N ]. Thus in general we have: We can then write the coset space as follows: where N ± = l ± ⊖l ± . Semisimplicity of g 3 N and g D N implies that dim(l + ) = dim(l − ) and dim(l + ) = dim(l − ), so that dim(N + ) = dim(N − ). More precisely, in a suitable basis, for each strictly-uppertriangular matrix M + representing an element in N + , its (strictly-lower-triangular) hermitian-conjugate M − = M † + = −τ (M + ) represents an element in N − : The former is given by a generator either of the form i (E α − σ(E α )) or E α + σ(E α ), for some α ∈∆ + [g 3 N ] ⊖∆ + [g D N ⊕ sl(D − 2)], the latter will either be −i (E −α − σ(E −α )) or E −α + σ(E −α ), corresponding to the same α. Thus if {L + ℓ }, ℓ = 1, . . . , dim(N ± ), is a basis of N + , {L − ℓ } = {−τ (L + ℓ )} is a basis of N − and we can also write the coset space in the form: B so (8,8) Outer Automorphisms and Dual Subalgebras of e 8(8) Consider in the maximal D = 3 theory the effect of an O (8,8) "reflection" of the form: where each block is an 8 × 8 matrix and D k is the zero-matrix except for only an odd number k of 1s along the diagonal. Such transformation, which belongs to the O(8) subgroup of O (8,8), is an outer automorphism of the D 8 algebra whose effect, modulo Weyl transformations of the same algebra, is to interchange α 2 with α 3 in Fig. 1. While it is a symmetry of the D 8 Dynkin diagram, it is not a symmetry of the e 8(8) one, as it changes the SO(8, 8)-chirality of the α 1 root, which is a D 8 -spinorial weight [23]. In particular this outer automorphism may map inequivalent subalgebras g, g ′ of so (8,8) into one another. This is the case of subalgebras g (and thus g ′ ) which are the direct sum of commuting A k -algebras with odd rank k. In mathematical language such dual subalgebras are said to be linearly equivalent, i.e. in any matrix representation they are equivalent through conjugation by means of a matrix, which is however not necessarily a representation of an SO (8,8) element, as it is the case for the outer automorphisms. Equivalence therefore implies linear equivalence though the reverse implication is not true. With respect to g and g ′ , a same spinorial representation of so (8,8), and thus the adjoint representation of the whole e 8(8) , will branch differently. They are clearly inequivalent e 8(8) -subalgebras. Examples are given in [38]: g = sl (8), sl(6) ⊕ sl (2), sl(4) ⊕ sl (4)  What has been said for so (8,8) also holds for the so * (16) and so(16) subalgebras of e 8 (8) . For instance there are two inequivalent u(8), u ′ (8) in either so * (16) or so (16). One contains the R-symmetry algebras su(8), usp (8), etc. of the higher dimensional parent maximal supergravities, the other dual subalgebras su ′ (8), usp ′ (8), etc. which are not contained in the chain of exceptional duality algebras e 7(7) , e 6(6) etc.
Let us briefly recall the relation between outer automorphisms of so (8,8) and dualities. Consider the toroidal reduction of the D = 11 theory down to D = 3 (in the Einstein frame). The Kaluza-Klein ansatz for the metric reads: whereμ,ν = 0, . . . , 10, µ, ν = 0, 1, 2, m, n = 3, . . . , 10 and the internal metric of T 8 is conveniently written as follows: where P µa b ≡ (Ê −1 ∂ µÊ ) a b and the dialtonic vector h has the following form in the (ǫ i ) orthonormal basis: The field strengths F µa and F µ ab are associated with the scalars χ n and χ mn dual in D = 3 to the vectors G m µ and A µmn respectively, while F µabc is the one pertaining to the scalars A mnp . In these conventions, the lower (or upper) internal SO(8)-indices a, b, c of these field strengths are related to the SL(8, R) indices m, n, p by means ofÊ (orÊ −1 ). For instance: The above Lagrangian can also be written in the more compact form: where the one-forms Φ (α) µ are associated with each of the e 8(8) -positive roots α [40,41,42]. 23 It is useful to express the various radial moduli σ a in terms of the corresponding fieldsσ a in the D = 10 string frame: we find: a ǫ a−2 + −2φ + 9 a=3σ a ǫ 8 . (B.9) 22 We adopt the mostly plus signature for the metric. 23 The representation (B.7) of the D = 3 Lagrangian applies to all D = 3 supergravities. In the general (non necessarily maximal) case, h is a suitable dilaton-dependent vector in the h nc subspace of the Cartan subalgebra of g 3 N , while α are the restrictions to h nc of the g 3 N positive roots (restricted roots, see [13]).
The outer automorphism O k in (B.1) has the effect of changing the sign to an odd number of ǫ a , or, equivalently, to their coefficients in h: We see that: . . , k. If i ℓ run between 1 and 7, this transformation amounts to a T -duality along the internal directions y i ℓ +2 [39,23]: These transformations map type IIA into type IIB theory. If k = 1 and i ℓ = 8 then there is an S-duality involved:σ ′ i =σ i and φ ′ = −φ + 9 a=3σ a . Instead of considering inequivalent T-dual subalgebras g, g ′ ⊂ so(8, 8) within a same e 8(8) algebra, we may adopt an equivalent point of view and consider a same subalgebra g ⊂ so (8,8) within two e 8(8) algebras, called in [23] e + 8 (8) and e − 8(8) , 24 defined respectively by completing the so(8, 8) Dynkin diagram with spinorial weights of different chiralities, namely attaching the weight α 1 to α 3 , as in Fig.  1, or a weight α ′ 1 to α 2 , defined as follows: This is useful if, for instance, we fix the g = gl(8, R) ⊂ so (8,8) group to be the same in the type IIA and type IIB settings. Then the different gl(8, R)-weights defining the dimensionally reduced type IIA and type IIB forms are obtained by branching the adjoint representations of e + 8 (8) and e − 8 (8) , respectively, with respect to the common gl(8, R), [23].
The doubling of the equivalence classes inside a D n algebra into dual pairs, discussed above, does not occur if the subalgebra is the sum of commuting algebras in the case in which either all of them are of type A k with even rank k, or at least one of them is of type D [36]. This is consistent with the fact observed in Subsect. 2.1, that the SL(7, R) D = 9 Ehlers subgroups of SO (8,8) which pertain to the type IIA and IIB descriptions are equivalent. The same rule guarantees that, in D = 6, the SO(5, 5) × SL(4, R) subgroups of SO (8,8) in the type IIA and IIB settings, are equivalent.

C Poincaré Duality and Level Decomposition
Consider now the branching of the adjoint representation of g 3 N with respect to SL(D − 2) × G D N : where it is understood that if (Λ d , R d ) = ( * Λ d , R ′ d ), they are counted just once in N d . In light of our discussion in Appendix A.3, we can write the coset space as the carrier of a representation d N d , namely rewrite eq. (A.26) as follows: In fact each subspace N d splits into conjugate nilpotent subalgebras as follows: this being a consequence of the property: τ (N d ) = N d . Each nilpotent subalgebra N + d or N − d separately defines a representation with respect to (the adjoint action of) G D N and the subgroup GL(D − 3) ⊂ SL(D − 2), though not with respect to SL(D − 2) itself. We can decompose each space N d into eigenspaces of the Cartan involution τ , consisting of compact and non-compact generators: These subspaces define representations with respect to the compact group SO(D − 2) × mcs(G D N ) and, moreover dim(N c d ) = dim(N nc d ) . (C.5) For the sake of simplicity, let us consider a split (maximally non-compact) g 3 N . Then each N d will be generated by shift operators corresponding to a certain set of positive roots α (d) and their negatives: , (C. 6) and the conjugate nilpotent subalgebras are N + d = Span(E α (d) ) and N − d = Span(E −α (d) ). The eigenspaces N nc d , N c d of the Cartan involution, consisting of compact and non-compact generators read: Each positive root α (d) corresponds to a D = 3 scalar field in the Lagrangian (B.7). For a given d the roots α (d) are defined by the level decomposition of the g 3 N -roots with respect to the root which is truncated out of its extended diagram in order to define the g D N ⊕ sl(D − 2)-subdiagram. 25 Let us illustrate this procedure in the maximal theory. As shown in Appendix A, the e 11−D(11−D) × sl(D − 2) diagram is obtained by deleting from the e 8(8) -extended Dynkin diagram the root α 12−D . The sl(D − 2) subalgebra is defined by the simple roots α 13−D , . . . , α 8 , −ψ, ψ = ǫ 1 + ǫ 8 being the e 8 (8) highest root, while its gl(D − 3) subalgebra only by the roots α 13−D , . . . , α 8 . Writing a generic e 8 (8) positive root in the simple root basis: the positive integer n i defines the level of α with respect to α i . Let us consider the level-decomposition with respect to the root α 12−D for dimensions D < 9, namely the values of n 12−D defining the roots α (d) . 26 25 In the non-split case, one should consider the level decomposition of the restricted roots. Level decompositions are a common procedure in the E10 and E11 approaches to maximal supergravity [43,44]. 26 More precisely, the level n i is the grading of the generator Eα with respect to the SO(1, 1) generator H λ i (i.e. [H λ i , Eα] = n i Eα), λ i being the g 3 N simple weights. The level decomposition is defined by the Cartan generator which is orthogonal to the Cartan subalgebra of g D N ⊕ gl(D − 3) (and therefore commutes with g D N ⊕ gl(D − 3)). In the maximal theory, for D < 9, the relevant Cartan generator is H λ 12−D and thus the level to consider is n12−D. For D = 9 the generator is H λ 2 + H λ 3 and so we shall consider the decomposition with respect to the integer n = n2 + n3. In the type IIA D = 10 description, the generator is H λ 1 +2 H λ 2 and the decomposition will be effected with respect to n = n1 +2 n2. D = 4. In the case of D = 4 we have 63 roots with n 8 = 0, corresponding to the e 7(7) -positive roots. The level n 8 = 1 roots are 56 and are the α (1) -roots whose shift generators E ±α (1) define the carrier space of the N d=1 = (1, 56) representation. The level n 8 = 2 root defines, with its negative, the shift generators in the quotient sl(D − 2) ⊖ gl(D − 3) = sl(2) ⊖ gl (1), which are the two shift generators of the Ehlers group. D = 5. Consider now the D = 5 case. There are 37 level-n 7 = 0 roots corresponding to the positive roots of e 6(6) ⊕ gl (2). The level-n 7 = 1 roots are 54 and define in N + d=1 a subspace in the (2, 27)-representation of SL(D − 3) × E 6(6) = SL(2) × E 6(6) , while the 27 level-n 7 = 2 roots define a subspace in the (1, 27 ′ ) with respect to the same group. The space N − d=1 will be the carrier of the conjugate representations. Together, the level n 7 = 1, 2 roots and their negatives define the space D = 9. The same analysis applies to D = 9, although in this case we shall consider the sum n = n 2 +n 3 . There are 16 roots with n = 0, which are the positive roots of the algebra gl(6)⊕gl (2). The α (1) roots consist of the 18 with n = 1 and the 3 with n = 6, defining the SL(6)×GL(2)-representations (6, 2 +3 + 1 −4 ) ⊕ (1, 2 −3 + 1 +4 ) in N + d=1 which, together with the conjugate representations in N − d=1 , complete the N d=1 = (7, 2 +3 + 1 −4 ) ⊕ (7 ′ , 2 −3 + 1 +4 ) of SL(7) × GL(2). The α (2) s are the 30 roots with n = 2 and the 12 with n = 5 defining the SL(6) × GL(2)-representations (15, 2 −1 ) ⊕ (6 ′ , 2 +1 ) in N + d=2 , so that N d=2 = N + d=2 ⊕ N − d=2 = (21, 2 −1 ) ⊕ (21 ′ , 2 +1 ). Next we have to consider the 20 n = 3 and the 15 n = 4 roots which make the α (3) and define the SL(6) × GL(2)-representations (20, 1 +2 ) ⊕ (15 ′ , 1 −2 ) in N + d=3 , so that N d=3 = N + d=3 ⊕ N − d=3 = (35, 1 +2 ) ⊕ (35 ′ , 1 −2 ). Finally the 6 n = 7 roots, with their negative, define the generators of the coset sl(D − 2) ⊖ gl(D − 3) = sl(7) ⊖ gl (6). D = 10, IIB. In the D = 10 case we have to distinguish between the type IIA and IIB theories. In the type IIB setting we need consider the level n 3 with respect to α 3 . The 22 roots with n 3 = 0 are the positive roots of gl(7) ⊕ sl(2), sl(2) being the U -duality group. In this case we only have d = 2, 4. The α (2) s consist of the 42 n 3 = 1 and the 14 n 3 = 3 defining the SL(7) × SL(2)-representations (21, 2) ⊕ (7 ′ , 2) in N + d=2 , so that N d=2 = N + d=2 ⊕ N − d=2 = (28, 2) ⊕ (28 ′ , 2) of SL(8) × SL(2). Next we have the 35 roots with n 3 = 2, which are the α (4) s and define the (35, 1) of SL(7) × SL (2) in N + d=4 , so that N d=4 = N + d=4 ⊕ N − d=4 = (70, 1). There are 7 n 3 = 4 roots which, with their negative, define the generators of the coset sl(D − 2) ⊖ gl(D − 3) = sl(8) ⊖ gl (7). D = 10, IIA. As far as the type IIA description is concerned, the level to consider for the decomposition is the sum n = n 1 + 2 n 2 . In this case we only have d = 1, 2, 3. There are 21 n = 0 roots which are the positive roots of gl(7) ⊕ so(1, 1), so(1, 1) being the U -duality algebra. The α (1) roots consist of the 7 roots with n = 1 and the single n = 7 root defining the SL(7)×SO(1, 1)-representation 7 +3 ⊕1 −3 in N + d=1 which, together with the conjugate representations in N − d=1 , complete the N d=1 = 8 +3 ⊕ 8 −3 of SL(8) × SO(1, 1). Next we consider the 21 roots with n = 2 and the 7 with n = 6, whose shift operators generating N + d=2 transform in the 21 −2 ⊕ 7 ′ +2 with respect to SL(7) × SO(1, 1). These roots define then the α (2) and N d=2 = N + d=2 ⊕ N − d=2 = 28 −2 ⊕ 28 ′ +2 of SL (8) (7). D = 11. We end this analysis with the D = 11 case discussed in the previous Section. The relevant level decomposition is with respect to the root α 2 . With n 2 = 0 we have the positive roots of gl (8).