A magic pyramid of supergravities

By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and single set of transformation rules, but with fields valued respectively in R,C,H,O, it was recently shown that tensoring left and right multiplets yields a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was subsequently tied in with the more familiar R,C,H,O description of spacetime to give a unified division-algebraic description of extended super Yang-Mills in D = 3, 4, 6, 10. Here, these constructions are brought together resulting in a magic pyramid of supergravities. The base of the pyramid in D = 3 is the known 4x4 magic square, while the higher levels are comprised of a 3x3 square in D = 4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10. The corresponding U-duality groups are given by a new algebraic structure, the magic pyramid formula, which may be regarded as being defined over three division algebras, one for spacetime and each of the left/right Yang-Mills multiplets. We also construct a conformal magic pyramid by tensoring conformal supermultiplets in D = 3, 4, 6. The missing entry in D = 10 is suggestive of an exotic theory with G/H duality structure F4(4)/Sp(3) x Sp(1).


Introduction
In recent years gauge and gravitational scattering amplitudes have undergone something of a renaissance [1], resulting not only in dramatic computational advances but also important conceptual insights. One such development, straddling both the technical and conceptual, is the colour-kinematic duality of gauge amplitudes introduced by Bern, Carrasco and Johansson [2]. Exploiting this duality it has been shown that gravitational amplitudes may be reconstructed using a double-copy of gauge amplitudes suggesting a possible interpretation of perturbative gravity as "the square of Yang-Mills" [3,4]. This perspective has proven itself remarkably effective, rendering possible previously intractable gravitational scattering amplitude calculations [5]; it is both conceptually suggestive and technically advantageous. Yet, the idea of gravity as the square of Yang-Mills is not specific to amplitudes, having appeared previously in a number of different, but sometimes related, contexts [6][7][8][9][10]. While it would seem there is now a growing web of relations connecting gravity to "gauge × gauge", it is as yet not clear to what extent gravity may be regarded as the square of Yang-Mills.
Here, we ask how the non-compact global symmetries of supergravity [11], or in an M-theory context the so-called U-dualities [12,13], might be related to the "square" of those in super Yang-Mills (SYM), namely R-symmetries. Surprisingly, in the course of addressing this question the division algebras A = R, C, H, O and their associated symmetries reveal themselves as playing an intriguing role. Tensoring, as in [14], N L and N R super Yang-Mills multiplets in D = 3, 4, 6, 10 dimensions yields supergravities with U-dualities given by a magic pyramid formula parametrized by a triple of division algebras (A n , A nN L , A nN R ), one for spacetime and two for the left/right Yang-Mills multiplets.
In previous work [15] we built a symmetric 4×4 array of three-dimensional supergravity multiplets, with N = N L + N R , by tensoring a left N L = 1, 2, 4, 8 SYM multiplet with a right N R = 1, 2, 4, 8 SYM multiplet. Remarkably, the corresponding U-dualities filled out the Freudenthal-Rosenfeld-Tits magic square [16][17][18][19][20][21]; a symmetric 4 × 4 array of Lie algebras defined by a single formula taking as its argument a pair of division algebras, (1.1) where the subscripts denote the dimension of the algebras. See Table 1. Here, tri(A) denotes the triality Lie algebra of A, a generalisation of the algebra of derivations which contains as a sub-algebra the R-symmetry of super Yang-Mills. See section 2.  The Freudenthal-Rosenfeld-Tits magic square 1 historically originated from efforts to understand the exceptional Lie groups in terms of octonionic geometries and, accordingly, the scalar fields of the corresponding supergravities parametrize division algebraic projective spaces [15]. The connection to the division algebras in fact goes deeper; the appearance of the magic square can be explained using the observation that the D = 3, N = 1, 2, 4, 8 Yang-Mills theories can be formulated with a single Lagrangian and a single set of transformation rules, using fields valued in R, C, H and O, respectively. Tensoring an A N L -valued super Yang-Mills multiplet with an A N L -valued super Yang-Mills multiplet yields a supergravity mulitplet with fields valued in A N L ⊗ A N L , making a magic square of U-dualities appear rather natural.
Of course, the connection between supersymmetry and division algebras is not new. In particular, Kugo and Townsend [26] related the existence of minimal N = 1 super Yang-Mills multiplets in only three, four, six and ten dimensions directly to the existence of only four division algebras R, C, H and O, an observation which has been subsequently developed in a variety of directions. See, for example, [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] and the references therein. From this point of view the division algebras are related to the spacetime symmetries, rather than the internal R-symmetries, via the Lie algebra isomorphism (in the sense of [27]) sl(2, A n ) ∼ = so(1, n + 2). (1.2) Indeed, the unique D = 10, N = 1 super Yang-Mills theory can be formulated using octonionic spacetime fields [39]. By dimensionally reducing this octonionic theory, which corresponds to Cayley-Dickson halving, one recovers the octonionic formulation of D = 3, N = 8 Yang-Mills presented in [15], tying together the division algebraic descriptions of spacetime and supersymmetry. This approach gives a unified division algebraic description of (D = 3, N = 1, 2, 4, 8), (D = 4, N = 1, 2, 4), (D = 6, N = 1, 2) and (D = 10, N = 1) Yang-Mills theories. A given (D = n + 2, N ) theory (the field content, Lagrangian and transformation rules) is completely specified by selecting an ordered pair A n ⊆ A nN , where again the subscripts denote the dimension of the algebras [41]. This unity is neatly expressed through the fact that the (modified) triality algebras appearing in (1.1) are the direct sum of the spacetime little group and the internal R-symmetry algebras [41].
In the present work we bring together the various roles of the division algebras discussed above to construct a pyramid of supergravities by tensoring left/right A nN L /A nN R -valued Yang-Mills multiplets in D = n + 2. The base of the pyramid in D = 3 is the 4 × 4 magic square of supergravities, with a 3 × 3 square in D = 4, a 2 × 2 square in D = 6 and Type II supergravity at the apex in D = 10. The totality defines a new algebraic structure: the magic pyramid. The U-dualities are given by the magic pyramid formula, There are a number of equivalent forms/constructions of the magic square formula (1.1) due to, amongst others, Tits [20], Vinberg [21], Kantor [22] and Barton-Sudbery [23]. The ternary algebra approach of [22] was generalised by  to include super Lie algebras. The form given in (1.1) is due to Barton-Sudbery. We adopt this construction, modified as a Lie algebra to produce the required real forms presented in Table 1, as will be explained in section 2. This specific square of real forms was first derived in [25] using Tits' formula defined over a Lorentzian Jordan algebra. By, for example, altering the signature of the algebras a variety of real forms can be accommodated. See [25] for a comprehensive account in the context of supergravity. where so(A n ) ST ⊂ L 3 (A nN L , A nN R ) is the subalgebra of on-shell spacetime transformations (the spacetime little group 2 in D = n + 2 dimensions is SO(n) ∼ = SO(A n ), as described in section 2). This is the natural generalisation of the magic square formula given in (1.1); the largest subalgebra of L 3 (A nN L , A nN R ) that respects spacetime transformations generates the U-duality. See Figure 1 for the magic pyramid of U-duality groups described by (1.3), and Figure 2 for the ranks of the corresponding cosets, which we include to highlight the the curious pattern they follow.
The pyramid formula may also be understood geometrically. As observed in [42] for the exceptional cases, the D = 3 Freudenthal magic square can be regarded as the isometries of the division algebraic projective spaces (A N L ⊗ A N R )P 2 . Here we are being rather heuristic -for more detailed and elegant treatments of magic square projective geometry see [34,[43][44][45] and the references therein. In essence the pyramid algebra describes the isometries of special submanifolds of these projective spaces. On tensoring SYM multiplets in D > 3, we must identify a diagonal A n subalgebra to be associated with spacetime. This can be thought of as introducing an A n -structure on the SO(N L + N R ) that acts (in the spinor representation) on the tangent space (A N L ⊗ A N R ) 2 at each point on the projective plane. This splits the projective space into two pieces, one internal and one spacetime. The isometries of the internal component yield the magic pyramid, while the remaining symmetries generate the spacetime little group.
Rather than SYM one might also consider "squaring" the D = 3, 4, 6 conformal multiplets: super Chern-Simons-matter (CSm), SYM and tensor, respectively [14]. This yields another magic pyramid, as described in section 4, which we will refer to as the conformal pyramid. See Figure 3. It has the remarkable property that its faces are also given by the known D = 3 magic square. For example, Figure 2: The ranks of the scalar cosets G/H, where G is the U-duality and H is its maximal compact subgroup (the entries here apply to the original magic pyramid obtained by squaring SYM rather that from squaring conformal theories).
trading the maximal super Yang-Mills in D = 6 for the (2, 0) tensor mulitplet swaps the resulting maximal supergravity with SO(5, 5) U-duality for the non-gravitational (4, 0) self-dual-Weyl multiplet with E 6(6) U-duality considered in [14,46,47]. Given the recent progress in understanding threedimensional supergravity amplitudes as double copies of Bagger-Lambert-Gustavsson theories [48], one might anticipate applications to this line of enquiry. More speculatively, the conformal pyramid in D = 3, 4, 6 suggests an exotic D = 10 theory with global symmetry F 4(4) , although it would have to be highly non-conventional (even heretical) from the standard perspective on the classification of supermultiplets. For earlier appearances of F 4 in 10 and 11 dimensions see [49][50][51][52][53].
Note, in the present paper the complete Freudenthal-Rosenfeld-Tits magic square describes the Udualities of conventional D = 3 supergravities. Its role here is not to be confused with its appearance in the important, and aptly named, "magic supergravities" of Günaydin-Sierre-Townsend [54,55]. In this context the C, H, and O rows of the magic square (with a different set of real forms) describe the U-dualities of the magic supergravities in D = 5, 4 and 3 respectively. The magic square also appeared previously in a further, distinct, supersymmetric setting in [56].
In section 2 we review division algebras and the square construction, giving the details of our formulation of Table 1, which were omitted from [15]. In section 3 we briefly recall the division algebraic description of SYM and then construct the magic pyramid of supergravitites. In section 4 we introduce the conformal pyramid.

The magic square
An algebra A defined over R with identity element e 0 , is said to be composition if it has a nondegenerate quadratic form 3 n : A → R such that, where we denote the multiplicative product of the algebra by juxtaposition. A composition algebra A is said to be division if it contains no zero divisors, in which case n is positive semi-definite and A is referred to as a normed division algebra. Hurwitz's celebrated theorem states that there are exactly four normed division algebras [57]: the reals, complexes, quaternions and octonions, denoted respectively by R, C, H and O.
Regarding R ⊂ A as the scalar multiples of the identity Re 0 we may decompose A into its "real" and "imaginary" parts A = R ⊕ A , where A ⊂ A is the subspace orthogonal to R. An arbitrary element a ∈ A may be written a = Re(a) + Im(a). Here Re(a) ∈ Re 0 , Im(a) ∈ A and Re(a) = 1 2 (a + a), where we have defined conjugation using the bilinear form, a := a, e 0 e 0 − a, a, b := n(a + b) − n(a) − n(b).
An element a ∈ O may be written a = a a e a , where a = 0, . . . , 7, a a ∈ R and {e a } is a basis with one real e 0 and seven e i , i = 1, . . . , 7, imaginary elements. The octonionic multiplication rule is, e a e b = (δ a0 δ bc + δ 0b δ ac − δ ab δ 0c + C abc ) e c , (2.4) where C abc is totally antisymmetric and C 0bc = 0. The non-zero C ijk are given by the Fano plane. See Figure 4. The norm preserving algebra is defined as, The triality algebra of A is defined as,  Given two normed division algebras A L and A R we can define on a Z 2 × Z 2 graded Lie algebra structure following, with a slight modification to accommodate the required real forms, Barton and Sudbery [23]. (2.11) (2.12) For two elements belonging to the same summand (A L ⊗ A R ) ij in (2.10) the commutator is defined using the natural map where the first arrow uses the norm on A L and A R . Explicitly, (2.14) Here T : With these commutators the magic square formula (2.10) describes the Lie algebras of the groups presented in Table 1. The simplest way to check that these definitions yield the correct non-compact real forms is by comparison to the conventions of Barton and Sudbery [23], which are known to give the magic square of compact real forms. Recall, a non-compact real form g nc of a complex semi-simple Lie algebra g C admits a symmetric decomposition g nc = h + p, where h is the maximal compact subalgebra. If a compact real form g c shares with some non-compact real form g nc a common subalgebra, g nc = h + p and g c = h + p , and the brackets in [h, p] are the same as those in [h, p ], but equivalent brackets in [p, p] and [p , p ] differ by a sign, then h is the maximal compact subalgebra of g nc . This observation is sufficient to confirm that our construction yields the real forms in Table 1 and we can identify as the maximal compact subalgebra. The corresponding compact subgroups are presented in Table 2. The key to constructing the magic pyramid is to write the appropriate Yang-Mills theories over R, C, H, O and then consider the symmetries of the "squared" theories. The Lagrangian for (n + 2)dimensional N = 1 SYM with gauge group G over the division algebra A n is given [39,41] by where the covariant derivative and field strength are given by the usual expressions The {σ µ } are a basis for A n -valued Hermitian matrices -the straightforward generalisation of the usual complex Pauli matrices [32,41,58] to all four normed division algebras, satisfying the usual Clifford algebra relations. We can use these to write the supersymmetry transformations: Note, since the components λ Aa are anti-commuting we are dealing with the algebra of octonions defined over the Grassmanns. By dimensionally reducing these theories using the Dixon-halving techniques of [41], we arrive at a master Lagrangian for SYM in D = n + 2 with N supersymmetries written over the division algebra A nN . The division algebra associated with spacetime A n is viewed as a subalgebra of A nN . The resulting Lagrangian is: where λ ∈ A 2 nN (so we have N spacetime spinors, each valued in A 2 n ) and φ is a scalar field taking values in φ ∈ A n , the subspace of A nN orthogonal to the A n subalgebra. The {σ µ } are still a basis for A n -valued Hermitian matrices, again, with A n viewed as a division subalgebra of A nN .
The supersymmetry transformations in this language are where the subscript A n refers to the projection onto this subspace and The important result is that we can write N = 1 SYM in D = n + 2 over the division algebra A n , and that if we wish to double the amount of supersymmetry we Dickson-double the division algebra. For example, if we start with N = 1 SYM in D = 4 over C, then N = 2 will be written over H and N = 4 will be written over O.
It is also useful to consider the little group representations of the fields in the division algebraic Yang-Mills theories. The little group in D = n + 2 dimensions is SO(n). When N = 1, SYM contains a single vector and its fermionic partner, which may each be represented as an element of A n , with the little group transforming them via the natural action of SO(A n ). For example, in D = 10 we have an octonion representing the vector, transforming as the 8 v of SO (8), while the spinor is represented by another octonion, transforming as the 8 s . As noted in [41], the overall (little group plus internal) symmetry of the N = 1 theory in D = n + 2 dimensions is tri(A n ). If we dimensionally reduce these theories we obtain SYM with N supersymmetries whose overall symmetries are given by where so(A n ) ST is the subalgbra of so(A nN ) that acts as orthogonal transformations on A n ⊂ A nN . The division algebras used in each dimension and the corresponding sym algebras are summarised in Table 3. The on-shell content of each SYM theory can then be summarised as: • an A nN element of bosons: a vector in A n and scalars in A n , giving A n ⊕ A n = A nN • an A nN element of fermions: N spacetime fermions each valued in A n . Yang-Mills theory. For example, one can see the familiar R-symmetries in D = 4: U(1), U(2) and SU(4) for N = 1, 2, 4, respectively. Note that the symmetries in D = 3 are entirely internal and that they include the R-symmetry as a subgroup (these are actually the symmetries of the theories after dualising the vector to a scalar, to be discussed in the following section).

The D = 3 magic square of supergravities
In D = 3 the spacetime algebra is A 1 = R, so the SYM master Lagrangian (3.4) describes N real two-component Majorana spinors λ α a , α = 1, 2, written as a single A N -valued object λ α , as well as an R-valued vector and an ImA N -valued scalar field. Tensoring the multiplets of left (L) and right (R) Yang-Mills theories, as in [15], we obtain the off-shell field content: In three dimensions the R-valued graviton and A N L ⊕ A N R -valued gravitino carry no degrees of freedom, but indicate that we have found the field content of a supergravity theory with N = N L +N R supersymmetries. On-shell the (A N L ⊗A N R ) 2 -valued scalar and Majorana spinor each have 2(N L ×N R ) degrees of freedom.  The maximal compact subgroups H of the magic square of U-dualities G are those given in the reduced magic square presented in Table 2. These are the largest linearly-realised global symmetries under which Ψ µ , ϕ and χ transform as a vector, spinor and conjugate spinor; We can better understand the origin of the magic D = 3 U-dualities if we consider the symmetries of their parent Yang-Mills theories. If we set the coupling constant g in (3.4) to zero then we may dualise the vector to a scalar and write the Lagrangian as: where φ and λ α each take values in A N (note that in D = 3 we do not need to take the real part of the fermion kinetic term since the sigma matrices are real and symmetric). The supersymmetry transformations become: Equation (3.9) and (3.10) enjoy a symmetry whose Lie algebra is sym(R, A N ) ∼ = tri(A N ). Looking at the theory it is clear why this is the case, since transformations preserving (3.10) coincide with the definition of tri(A N ); that is to say, we might initially try to transform the three A N -valued objects φ, λ and by three independent SO(A N ) rotations, but the supersymmetry transformations constrain these three rotations to satisfy the definition (2.6).
Tensoring the on-shell field content, φ L/R , λ L/R , of the dualised Yang-Mills theories we obtain an (A N L ⊗ A N R ) 2 -valued scalar and spinor, The tri(A N L ) and tri(A N R ) symmetries of the two Yang-Mills theories act on these doublets as the (operator valued) 2 × 2 matrices: Since we have arranged the fields of the squared theory into doublets we might also consider the off-diagonal rotations 0 e * a ⊗ e * a −e a ⊗ e a 0 . (3.13) The total algebra of linear transformations is then given by which is precisely that of the reduced magic square. It is interesting to note that the off-diagonal transformations of (3.13) take Yang-Mills fermions into Yang-Mills bosons, and vice versa, but are bosonic generators in the supergravity theory. It is tempting to identify these generators with Q L ⊗ Q R , where Q L/R are the supercharges of the left and right Yang-Mills theories, as this correctly reproduces basic structure of transformations (3.13). However, the derivative in the supersymmetry transformations appears to spoil this correspondence. The full non-linear U-duality groups G are fixed by the field content and H symmetries, as described in [42]. The groups we find are, of course, those of the magic square in Table 1. We summarise the theories of the D = 3 magic square in Table 4.

Generalisation to D = 4, 6, 10
To generalise the results above to D = 4, 6, 10 we first consider the simplest example in D = 4: tensoring two N = 1 SYM multiplets (A µ , λ) to obtain N = 2 supergravity. Counting the degrees of freedom (4 × 4 = 16) tells us that we must have a gravity multiplet coupled to one hypermultiplet, so the field content we expect is: (g µν , 2Ψ µ , A µ , 2χ, 4φ). We will square on-shell, so the Yang-Mills fields are represented by the complex numbers (helicity states): Squaring and arranging into doublets of bosons and fermions then gives us the (C⊗C) 2 valued objects: Consider acting on these with (2) is given by the matrices 18) and the C L ⊗ C R part consists of the four anti-Hermitian matrices When working with C ⊗ C it is convenient to define the quantities which each seperately behave like the basis of C: but annihilate one another 4 : Rewriting the 8 matrices above in terms of this basis we obtain the following set: where the sigmas and epsilon refer to the usual Pauli matrices (as opposed to the generalised Pauli matrices defined previously): This set of matrices evidently generates SU (2) (2), as stated in the magic square in Table 2. However, the original Yang-Mills theories have a symmetry under the transformations where θ ST is the spacetime U(1) ST little group parameter and θ L/R I are those of the Yang-Mills Rsymmetries. Note that these correspond to tri(C L/R ) ∼ = so(2) ⊕ so (2). The variation of the fermionic doublet under U(1) ST and the R-symmetries is thus Written in terms of the i ± basis we find What has emerged are the U(1) ST transformations for the gravitinos and spin-1 2 fermions and two internal U(1) pieces. Since C ⊗ C is not a division algebra, it contains zero divisors; it is interesting to note their role here in ensuring that the spin-3 2 and spin-1 2 fields each receive their appropriate little group transformations. Now the largest symmetry that acts on this doublet must be the subalgebra of L 1 (C L , C R ) that commutes with these spacetime generators. All the matrices commute with i + 1, but i − σ 1 and 1 − ε do not commute with i − σ 3 , so we are forced to discard these generators. The remaining matrices generate: where the subscript U denotes the maximal compact subgroup of the U-duality. This is the entry found in the pyramid. Again, note how the gravitino transforms as a doublet under the SU(2) but, because of the i + , the spin-1 2 fields are singlets, as required in the supergravity theory. The Yang-Mills R-symmetries have been absorbed into the U-duality group. A similar analysis for the bosonic fields in the theory shows that we do indeed obtain a graviton, a vector and two scalars, which transform as a singlet, a singlet and doublet under the SU(2), just as one would hope.
In the general case for D = n + 2, we begin with a pair of Yang-Mills theories with N L and N R supersymmetries written over the division algebras A nN L and A nN R , respectively. Taking the little group fields we may then write all the bosons of the left (right) theory as a single element B L ∈ A nN L (B R ∈ A nN R ), and similarly for the fermions F L ∈ A nN L (F R ∈ A nN R ). After tensoring we arrange the resulting supergravity fields into a bosonic doublet and a fermionic doublet, just as we did in D = 3. The largest algebra that can act on these doublets is L 1 (A nN L , A nN R ), but an so(A n ) ST subalgebra of this corresponds to spacetime transformations, so we must restrict Similarly, for the full non-compact groups G it is, of course, a necessary condition that the required subalgebra of L 3 (A nN L , A nN R ) commutes with so(A n ) ST . Imposing this condition actually turns out to be sufficient to find all the correct U-dualities. The Lie algebra u of the U-duality group of a D = n + 2 supergravity theory obtained by tensoring Yang-Mills theories with N L and N R supersymmetries is thus given by: Locating the spacetime subgroup in the supergravity just amounts to tracking it back to the Yang-Mills theories by decomposing tri(A nN L/R ) into sym(A n , A nN L/R ). The systematic process for finding the U-dualities can then be summarised by the following recipe: • Discard all generators that transform non-trivially under SO(A n ) ST .

(3.42)
Note that the generators in the first pair of square brackets belong to the maximal compact subgroup SO(5) × SO(5), and those in the second pair are all non-compact, so we do indeed find the maximally non-compact real form SO (5,5), familiar from the dimensional reduction of Type II supergravity on T 4 .
Although we do not consider them directly here, it should be noted that the magic D = 6, N = (1, 0) supergravities (which come coupled to 2, 3, 5, 9 tensor multiplets and 2, 4, 8, 16 vector multiplets, respectively, as well as hypers) are closely related to the magic square and constitute the parent theories of the magic D = 5, 4, 3 supergavities. See [60] and the references therein.
where, once again, we use square brackets to divide the generators into those that belong to the maximal compact subgroup SO (16) and those that do not. We should again take the diagonal subgroup in SO(8) ST L × SO(8) ST R , taking tensor products of the representations appearing in the two slots: Discarding all but the spacetime singlets leaves us with a copy of SL(2, R) decomposed into the trivial group, which becomes leaving a single non-compact 1 to generate O(1, 1). This is the correct U-duality, since there is only a single scalar in Type IIA, which lives on the scalar manifold R ∼ = O(1, 1)/1.

Complex and Quaternionic Structures
It is interesting to look at the magic pyramid of maximal compact subgroups, shown in Figure 5.
The striking feature is that the D = 3 square is built from orthogonal groups, the D = 4 square from unitary groups and the D = 6 square from symplectic groups. This is no mere coincidence: SO(N ) is the group of rotations in a real N -dimensional space, U(N ) is the group of rotations in a complex N -dimensional space and Sp(N ) is the group of rotations in a quaternionic N -dimensional space [34,61], where the U(1) factor of U(N ) ∼ = SU(N ) × U(1) is generated by the complex structure J itself. In our case the complex structure is the generator of the spacetime little group U(1) ST .
To understand the different R-symmetry groups as we ascend from D = 3 to D = 6 we require the notion of a quaternionic structure. This is a triple of 4N × 4N matrices J 1 , J 2 and J 3 := J 1 J 2 satisfying the quaternion algebra: which we find belong to the Lie algebra so(4N ). Just as U(N ) may be seen as the subgroup of SO(2N ) that commutes with a complex structure, symplectic groups are the subgroups of SO(4N ) that commute with the J i : (note that the conditions on J 1,2 in the curly brackets are enough to ensure that (3.50) is satisfied). Just as the complex structure J generates u(1), the quaternionic structure matrices J i themselves generate a copy of sp(1), on account of (3.50), which by construction commutes with the sp(N ). In D = 6 the spacetime little group is SO(4) ST ∼ = Sp(1) + ST × Sp(1) − ST , and we can understand each of these Sp(1) factors as being generated by a quaternionic structure. For example, in the maximal (N + , N − ) = (2, 2) supergravity we have 16 supercharges, divided equally between the two chiralities, splitting the possible SO(16) group of transformations into SO(8)×SO (8). Putting a quaternionic structure on each of these SO(8) factors leaves us with an overall symmetry Sp(2) × Sp(2) × Sp(1)

S-duality of N = 4 SYM and supergravity
When the tensor product involves at least one N = 4 SYM multiplet, it is tempting to speculate that the exact SL(2, Z) S-duality of N = 4 SYM contributes to the S-duality of the resulting supergravity. How this might actually work remains unclear, especially given the exchange of the gauge group for its GNO (Goddard, Nuyts, and Olive) dual 7 [62]. However, a minimal consistency requirement can be checked. The SL(2, Z) S S-duality of supergravity acts nontrivially on the NS-NS sector gauge potentials and their duals, which together transform as doublets. The RR sector potentials and their duals are, on the other hand, singlets. The NS-NS potentials are identified as those originating from φ L ⊗ A Rν and A Lµ ⊗ φ R products, while the RR sector potentials come from spinor-spinor products λ L ⊗ λ R (consistent with the familiar type II story in D = 10). This yields the following counting of NS-NS and RR potentials and dual potentials, Decomposing the U-duality representations carried by the gauge potentials and their duals under the product of S and T dualities we have, demonstrating a splitting of the potentials into their NS-NS and RR sectors consistent with their tensor origin. Note, the SL(2, R) factor appearing in the U-duality of [N = 2] L ⊗ [N = 2] R , which yields N = 4 supergravity coupled to two vector multiplets, is not, as one might naturally assume, the S-duality group since it mixes NS-NS and RR, as can be checked by regarding it as a consistent truncation of N = 8 supergravity. However, the SL(2, R) S-duality inside E 7(7) is retained inside the SO(6, 2) factor of the N = 4 theory since SL(2, R) × SO(6, 2) is not a subgroup of SO (6,6). Of course, the strong-weak dualities of N = 2 SYM theories are not exact 8 and, as such, their role in this context is even less clear.

The conformal magic pyramid
Rather than uniformly tensoring SYM in each dimension we may consider instead the conformal theories: CSm in D = 3, SYM in D = 4 and tensor multiplets in D = 6. It is not clear what the appropriate theory should be in D = 10 and we leave this question for future work.
The tensorings of CSm and SYM in D = 3 yield the same results so it is only the tensor multiplets in D = 6 that we need to treat here. As for left/right SYM, composing tensor multiplets with opposing chiralities we obtain pure supergravity, reproducing Table 6.
On the other hand, for left/right tensor multiplets with matching chiralities we obtain the exotic non-gravitational SD-Weyl (self-dual-Weyl) multiplets coupled to tensor multipets, The [(2, 0) L tensor ] × [(2, 0) R tensor ] = [(4, 0) SD-Weyl ] squaring is given explicitly in Table 12. This theory is developed in some detail in [46,47]. It is non-gravitational with highest spin field transforming as the (5, 1) of the little group Sp(1) + ST × Sp(1) − ST . The terminology "SD-Weyl" derives from the fact that the (5, 1) representation has the symmetry properties of a four-dimensional Euclidean self-dual Weyl tensor when written with SO(4) indices, as described in [46].
The scalars of the SD-Weyl multiplets appearing in (4.2) parameterise the following cosets [(4, 0) SD-Weyl ] E 6(6) Sp(4) , Hence, by exchanging SYM multiplets with tensor multiplets the D = 6 level of the pyramid is adjusted, while the remaining levels are left unchanged. Interestingly, this has the consequence that the exterior faces of the pyramid, as presented in Figure 3, are given by the original magic square cut across its diagonal. An intriguing property of the tensor multiplets and SD-Weyl multiplets above is that every field is a scalar under Sp(1) − ST , so the spacetime symmetry is essentially just Sp(1) + ST as long as we multiply tensor multiplets of a single chirality 9 . Mathematically the conformal pyramid is perhaps the most natural, since we can understand: • the D = 4 square as the Freudenthal-Rosenfeld-Tits magic square restricted to the subgroups that commute with a complex structure • the D = 6 square as the Freudenthal-Rosenfeld-Tits magic square restricted to the subgroups that commute with a single quaternionic structure (as opposed to the pair of quaternionic structures we found for the SYM-squared pyramid).
See Figure 6. From this perspective a method for obtaining the (1, 0) tensor multiplet from the (1, 0) Yang-Mills multiplet would be to identify Sp (1) (2). The U-dualities u of the conformal pyramid are then given by In practice we find the groups of the conformal pyramid (in D = 3, 4, 6) using the following method: • Identify g ST as the diagonal subalgebra of g ST L ⊕ g ST R , • Discard all generators that transform non-trivially under the spacetime symmetries g ST .
Once again, to find the maximal compact subgroups we just replace in the above. While this method does not tell us how to obtain the U-duality in D = 10, we venture some speculations on this matter as part of our closing remarks in section 5.  is solved by where The δ 2n u(1) ST L/R terms come from the fact that when n = 2, g ST ∼ = u(1) and so commutes with itself. The group we identify as spacetime in the supergravity theory is the diagonal subgroup of the left and right spacetime groups; subtracting this we are left with Finally we denote the solution to since its dimension is nN L N R , and this notation captures its essence; we have made A nN L/R look like ∼ A n A N L/R , and then brought the left and right pieces together, identifying a diagonal A n algebra. Putting all of this together we arrive at a Barton-Sudbery-style formula for the compact subgroups of the conformal pyramid which allows one to build up the symmetries of the squared theories from those of the left/right conformal theories.

Conclusions
We began with the observation developed in [41] that N = 2 m -extended SYM theories in D = n + 2 spacetime dimensions may be formulated with a single Lagrangian and single set of transformation rules, but with spacetime fields valued in A nN . This perspective reveals a role for the triality algebras; once the fields are regarded as division algebras consistency with supersymmetry constrains the possible space of transformations to sym(A n , A nN ) ⊆ tri(A nN ). Tensoring left/right SYM multiplets valued in A nN L/R then naturally leads us to N L + N R supergravity multiplets with spacetime fields valued in A nN L ⊗ A nN R . For D = 1 + 2 this yields a set of supergravities with U-duality groups given by the magic square of Freudenthal-Rosenfeld-Tits. For n = 2, 4, 8, identifying a common spacetime subalgebra A n truncates the magic square to a 3×3, 2×2, and 1 × 1 array of subalgebras, corresponding precisely to the U-dualities obtained by tensoring SYM in D = 4, 6 and 10, respectively. Together the four ascending squares constitute a magic pyramid of algebras defined by the magic pyramid formula (3.30). The exceptional octonionic row and column of each level is constrained by supersymmetry to give the unique supergravity multiplet. On the other hand, the interior 3 × 3, 2 × 2, 1 × 1 and 0 × 0 squares can and do admit matter couplings. These additional matter multiplets are just as required to give the U-dualities predicted by the pyramid formula. Interestingly, in these cases the degrees of freedom are split evenly between the graviton multiplet and the matter multiplets, the number of which is determined by the rule 10 k = min(N L , N R ).
The magic pyramid supergravity theories are rather non-generic. Not only are they, in a sense, defined by the magic pyramid formula, they are also generated by tensoring the division algebraic SYM multiplets. It would therefore be interesting to explore whether they collectively possess other special properties, particularly as quantum theories, which can be traced back to their magic square origins. For example, in the maximal [N L = 4 SYM] × [N R = 4 SYM] case it has been shown that N = 8 supergravity is four-loop finite [5], a result which cannot be attributed to supersymmetry alone. While N = 8 is expected to have the best possible UV behaviour, as suggested by its connection to N = 4 SYM, it could still be that the remaining magic square supergravities share some structural features due to their common gauge × gauge origin and closely related global symmetries.
Conversely, one might also seek extensions of the magic pyramid construction which could account for more generic supergravities. The magic supergravities of Gunaydin, Sierre and Townsend [54,55], for example, admit at least one obvious generalisation using the family of spin-factor Jordan algebras, suggesting a possible extension of the present construction by incorporating matter multiplets.
Let us return to the present treatment, now in the conformal case. In section 4 we saw that tensoring the conformal theories in D = 3, 4, 6 resulted in a pyramid with the intriguing feature that its exterior faces are given by the Freudenthal-Rosenfeld-Tits square cut across its diagonal. In particular, ascending up the maximal spine one encounters the famous exceptional sequence E 8(8) , E 7(7) , E 6(6) , but where E 6(6) belongs to the exotic (4, 0) theory in D = 6. This pattern suggests the existence of some highly exotic D = 10 theory with F 4(4) U-duality group 11 . We would naturally require it to dimensionally reduce to the (4, 0) theory in D = 6 on some non-trivial manifold (orbifold), consistent with scalars living in E 6(6) / Sp(4) and F 4(4) / Sp(3) × Sp(1) in D = 6 and 10, respectively. The D = 6 supercharges transform as the 8 of Sp (4), which breaks to the (6, 1) + (1, 2) of Sp(3) × Sp(1), leaving N = 2, 6, 8 as possibilities in D = 10. Naively at least, N = 2 is ruled out by the standard classification of supermultiplets [64,65] due to the assumption that it may be dimensionally reduced to D = 6, N = (4, 0), since this would imply fields of helicity greater than 2 when dimensionally reducing on a 6-torus. If, however, the F 4(4) theory had some exotic dynamics which broke the usual spacetime little group to some subgroup this logic may not hold. Taking into account the desired F 4(4) , this line of reasoning suggests a G 2 little group as one possibility. These avenues will be explored elsewhere.
There is, however, an obvious alternative interpretation of the conformal pyramid including its F 4(4) tip. The (4, 0) theory in D = 6 with E 6(6) U-duality reduces on a circle to D = 5, N = 8 supergravity, again with E 6(6) U-duality. The same result holds for the remaining three slots of the D = 6 square; they each reduce to a D = 5 supergravity theory with very same U-duality group. This is a consequence of the fact that the D = 6 fields are singlets under the second factor of the little group Sp(1) × Sp(1), which therefore effectively reduces to the D = 5 little group Sp (1). Each of the resulting D = 5 theories may be obtained by squaring. Hence, Figure 3 may be regarded as a squashed pyramid of U-dualities for theories in D = 3, 4, 5, 6. The apex is now given by the D = 6, N = (3, 1) theory, obtained from [(2, 0) L tensor ] × [(1, 1) R vector ], with G/H given by F 4(4) / Sp(3) × Sp(1), as expected. Note, however, this multiplet contains gravitini but no graviton and is therefore not expected to define a consistent interacting theory.
We conclude with some brief remarks on the geometrical interpretation of the magic pyramid. When we made the observation that the U-dualities of the magic pyramid could be regarded as the isometries of the Lorentzian projective planes (A N L ⊗ A N R )P 2 (or submanifolds thereof), this was meant rather loosely in  (2) projection operators cannot be made [34]. Nonetheless, they are in fact geometric spaces, generalising projective spaces, known as "buildings", on which the U-dualities act as isometries. Buildings where originally introduced by Jacques Tits to provide a geometric approach to simple Lie groups, in particular the exceptional cases, but have since had far reaching implications. See, for example, [66,67] and the references therein. Of course, it has long been known that increasing supersymmetry restricts the spaces on which the scalar fields may live, as comprehensively demonstrated for D = 3 in [42]. Here we see that these restrictions lead us to the concept of buildings. It may be of interest to examine whether this relationship between supersymmetry and buildings has some useful implications.
A D = 6 Tensoring tables In Table 8, Table 9 and Table 11 we perform the D = 6 SYM squaring on-shell to arrive at the supergravity and matter content. In each table the fields are shown together with their little group SO(4) ∼ = Sp(1) + × Sp(1) − representations. Note, we restrict to the semi-simple part of the full little group for massless states. The [(2, 0) L tensor ] × [(2, 0) R tensor ] = [(4, 0) SD-Weyl ] tensoring is given as an example in Table 12. The little group representations appearing in the left-handed SD-Weyl multiplets are given by (n, 1), where n = 1, 2, . . . 5. These irreps are carried by totally symmetric rank n − 1 tensors of Sp(1) + : The multiplicities are given by the dimension of the R-symmetry representation of the fields. Consulting Table 12 we see that there are 27 self-dual two-form field strengths transforming as the fundamental 27 of E 6(6) . There are 42 scalars parametrising E 6(6) / Sp(4). The fermonic fields, C + and λ + , transform as the 8 and 48 of Sp(4) respectively.