Global symmetries of Yang-Mills squared in various dimensions

Tensoring two on-shell super Yang-Mills multiplets in dimensions D ≤ 10 yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{D} $$\end{document} with each dimension 3 ≤ D ≤ 10 we obtain a formula for the supergravity U-duality G and its maximal compact subgroup H in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.


Introduction
The idea of understanding aspects of quantum gravity in terms of a double-copy of gauge theories has a long history going back at least to the Kawai-Lewellen-Tye relations of string theory [1]. There has since been a wealth of developments expanding on this concept, perhaps most notably, but certainly not exclusively, in the context of gravitational and gauge scattering amplitudes. See for example . Indeed, invoking the Bern-Carrasco-Johansson colour-kinematic duality it has been conjectured [8] that the on-massshell momentum-space scattering amplitudes for gravity are the "double-copy" of gluon scattering amplitudes in Yang-Mills theory to all orders in perturbation theory. This remarkable and somewhat surprising proposal motivates the question: to what extent can one regard quantum gravity as the double copy of Yang-Mills theory? In this context it is natural to ask how the symmetries of each theory are related. In recent work [26] it was shown that the off-shell local transformation rules of (super)gravity (namely general covariance, local Lorentz invariance, p-form gauge invariance and local supersymmetry) may be derived from those of flat space Yang-Mills (namely local gauge invariance and global super-Poincare) at the linearised level.
Equally important in the context of M-theory are the non-compact global symmetries of supergravity [28], which are intimately related to the concept of U-duality [29,30]. For previous work on global symmetries in D = 4 spacetime dimensions via squaring see [6,14,15,24]. It was shown in [31] that tensoring two D = 3, N = 1, 2, 4, 8 super Yang-Mills mulitplets results in a "Freudenthal magic square of supergravity theories", as summarised in table 1. The corresponding Lie algebras of table 1 are concisely summarised by the magic square formula [31,32],  [34][35][36][37][38]. f denotes the total number of degrees of freedom in the resulting supergravity and matter multiplets.
which takes as its argument a pair of division algebras A N L , A N R = R, C, H, O, where we have adopted the convention that dim A N = N . The triality algebra of A, denoted tri(A), is related to the total on-shell global symmetries of the associated super Yang-Mills theory [33]. This rather surprising connection, relating the magic square of Lie algebras to the square of super Yang-Mills, can be attributed to the existence of a unified This observation was subsequently generalised to D = 3, 4, 6 and 10 dimensions [33,39] by incorporating the well-known relationship between the existence of minimal super Yang-Mills theories in D = 3, 4, 6, 10 and the existence of the four division algebras R, C, H, O [40][41][42][43][44]. From this perspective the D = 3 magic square forms the base of a "magic pyramid" of supergravities.These constructions build on a long line of work relating division algebras and magic squares to spacetime and supersymmetry. See  for a glimpse of the relevant literature. An early example, 1 closely related to the present contribution, appears in work the Julia [49] on group disintergrations in supergravity. The oxidation of N -extended D = 3 dimensional supergravity theories yields a partially symmetric "trapezoid" of non-compact global symmetries for D = 3, 4, . . . 11 and 0, 2 0 , 2 1 , . . . 2 7 supercharges. 2 The subset of algebras in the trapezoid given by D = 3, 4, 5 and 2 5 , 2 6 , 2 7 supercharges fits into the 3 × 3 inner C, H, O part of the magic square, excluding the (C, C) entry. Note, the exact symmetry of this subsquare is broken by the precise set of 1 As far as we are aware the first instance in this context. 2 It also includes the affine Kac-Moody algebras, e9 = e + 8 , e + 7 , e + 6 , so + 10 in D = 2 as made more precise in [86].

JHEP01(2016)148
real forms obtained, which are not given by any magic square formula in the conventional sense. 3 However, this set of theories also matches the D = 3, 4, 5 exterior wall of the pyramid in figure 2 obtained by squaring Yang-Mills and the corresponding algebras are indeed given by the pyramid formula (1.4) described in this work. Note, dispensing with the requirement of supersymmetry the same 3 × 3 square but with maximally non-compact real forms was derived as a corner of a "magic triangle" of theories in 3 ≤ D ≤ 11 spacetime dimensions [66]. The entries of triangle are parametrised by the dimension D of the theory and the rank 0 ≤ n ≤ 8 of its symmetry algebra. The complete magic triangle displays a remarkable symmetry under D → 11 − n, n → 11 − D. It should be noted that these are not the only magic triangles of Lie algebras, a particularly elegant and intriguing example being that of Cvitanović [88,89].
Returning to the theme of gravity as the square of Yang-Mills, the magic pyramid of [39] corresponds to a rather special subset of supergravity theories: those given by tensoring the D = 3, 4, 6, 10 division algebraic super Yang-Mills theories constructed in [33]. In the present work we address the natural question of generalisation beyond this select subclass of theories and give a new formula which makes the double-copy structure manifest.
In section 2 we consider all tensor products of left N L -extended and right N R -extended super Yang-Mills multiplets in D = 3, . . . , 10 dimensions and introduce three formulae describing the global symmetries of the resulting (N L +N R )-extended supergravity multiplets: (1.2) 2. The algebra h(N L + N R , D) of H, the maximal compact subgroup of the U-duality group G, 3. The algebra g(N L + N R , D) of the U-duality group G itself, Here we have used ⊕ and + to distinguish the direct sum between Lie algebras and vector spaces; only if [m, n] = 0 do we use m ⊕ n. The meaning of these formulae and, in particular, their relation to the symmetries of the left and right super Yang-Mills factors, will be described in section 2. For the moment we simply note that they make the left/right structure manifest and uniform for all N L , N R and D and, as we shall see, each summand appearing in the three formulae has a natural lef t ⊗ right origin. The groups H and G corresponding to (1.3) and (1.4) are given in the generalised pyramids of figure 1 and figure 2, respectively. For these groups, the formulae presented above can be regarded as generalised "matrix models", in the sense of [32] (not to be confused with (M)atrix models), for classical and exceptional Lie algebras. As a matrix model, it is perhaps not as elegant

JHEP01(2016)148
as those presented in [32]. For one, we make no use of the octonions. However, it has the advantage, from our perspective, that it describes systematically all groups obtained by squaring super Yang-Mills and, moreover, makes the left and right factors manifest.
For N L + N R half-maximal or less the super Yang-Mills tensor products yield supergravity multiplets together with additional matter multiplets, as described in table 2. They may always be obtained as consistent truncations or, in many cases, factorised orbifold truncations of the maximally supersymmetric cases, as in [24]. The type, number and coupling of these multiplets is fixed with respect to (1.3) and (1.4). However, as we shall describe in section 3.1, by including a non-supersymmetric factor in the tensor product these matter couplings may be generalised to include an arbitrary number of vector multiplets (thus clearly not truncations). This procedure naturally yields analogous formulae for h and g, corresponding to specific couplings. The nature of these couplings is in a certain sense as simple as possible. This follows from the symmetries, which may be regarded as a consequence of simple interactions, assumed to be present in the non-supersymmetric factor of the tensor product.
where a, i and a , i are indices of the appropriate int(N L , D) and int(N R , D) representations, respectively. Note, we will always dualise p-forms to their lowest possible rank consistent with their little group representations, for example, B µν → φ, A µ in D = 4, 5,

JHEP01(2016)148
respectively. This ensures U-duality is manifest. The particular set of Ramond-Ramond p-forms φ aa RR + · · · one obtains is dimension dependent. The detailed form of these tensor products for D > 3 are summarised in table 2 and  table 3  (2.5)

R-symmetry algebras
We begin with the simple relationship between the R-symmetry algebras of supergravity and its generating super Yang-Mills factors. While somewhat trivial this example introduces much of the notation and concepts needed later for the H and G algebras. R-symmetry is defined here as the automorphism group of the supersymmetry algebra. Its action on the N -extended supersymmetry generators Q is given schematically by The R-symmetry algebra is fixed by the reality properties of the minimal spinor representation in D mod 8 dimensions. See, for example, [90]. Making use of the super-Jacobi identities, it can be shown that the U A 's form a representation of the algebra so(N ), u(N ), sp(N ) for Q real, complex, quaternionic (pseudoreal), respectively. For s − t = 2, 3, . . . 10 mod 8, where D = t + s, the spinor representations follow the famous Bott periodic sequence C, H, H⊕H, H, C, R, R⊕R, R, C, H, H⊕H, H, . . . Since R-symmetry commutes with the Lorentz algebra, only the reality properties of the spinor representation and N are relevant. Consequently, we may associate a (direct sum of) division algebra(s), denoted D, to every dimension, as given in table 4, which will then JHEP01(2016)148 +c.c.

JHEP01(2016)148
Let us briefly review these ideas here. For a detailed survey see [43,91]  (2.7) Using the standard matrix commutator these constitute the classical Lie algebras

JHEP01(2016)148
Let sa(n, A) denote their special subalgebras: (2.9) The seemingly undemocratic definition of sa(n, A) follows naturally from the geometry of projective spaces since Isom(AP n−1 ) ∼ = sa(n, A) (2.10) for A = R, C, H. In the octonionic case only OP 1 and OP 2 constitute projective spaces with Isom(OP 1 ) ∼ = so (8) and Isom(OP 2 ) ∼ = f 4(−52) , reflecting their exceptional status. It then follows that the N -extended R-symmetry algebras in D dimensions, denoted ra(N , D), are given by ra(N , D) = a(N , D), (2.11) where for N = (N + , N − ), as is the case for D = 6, 10, we have used the definition Since D[m, n] ∼ = D m ⊗ D n forms a natural (but not necessarily irreducible) representation of a(m, D) ⊕ a(n, D), it follows quite simply that the (N L + N R )-extended R-symmetry algebra is given by the N L and N R R-symmetry algebras via (2.13) The commutators for elements (2.14) These commutation relations follow from the standard matrix commutators of . The formula (2.13) and its commutators (2.14) amount to the well-known statement that the pairs [so(p + q), so(p) ⊕ so(q)], [su(p + q), su(p) ⊕ su(q) ⊕ u(1)] and [sp(p + q), sp(p) ⊕ sp(q)] constitute type I symmetric spaces. From the perspective of the left/right tensor product, a(N L , D)⊕a(N R , D) is generated directly by the R-symmetries of the left and right factors acting on Q L and Q R independently. However, together they form an irreducible doublet (Q L , Q R ) ∈ D N L ⊕ D N R (suppressing the spacetime little group spinor representation space), which must be rotated by , which completes (2.13) as is made clear by (2.15). In the sense to be described in section 2.3, these additional elements can be generated by Q L ⊗ Q R ∈ D[N L , N R ] by formally neglecting its little group representation space.
It follows from [31] that for D = 3 the R-symmetry algebras admit an alternative Freudenthal magic square description. Recall that the U-duality groups in D = 3 form the Freudenthal Magic square given by where der denotes the derivation algebra, J 3 (A) is the Jordan algebra of 3 × 3 Hermitian matrices over A and J 0 3 (A) is its traceless subspace. See for example [43]. One can generalise this construction for any rank of the Jordan algebra J n (A), where for n > 3 we must exclude the octonionic case [32]. The supergravity R-symmetry algebras in D = 3 are given by n = 2,

H algebras
and are uncharged under the remaining factors. The corresponding pyramid of H groups, which generalises the magic H pyramid of [39], is presented in figure 1.  supergravity states [4,6,26]. Seeking, instead, internal bosonic transformations on the supergavity multiplet that follow from supersymmetry on the left and right Yang-Mills multiplets suggests starting from the rather unconventional tensor product of the left and right supercharges, Q ⊗Q. That this might work, at least formally, follows from the observation where we are explicitly suppressing the spacetime indices.
Adopting the spinor-helicity formalism, a simple concrete example sufficient to illustrate the principle is given by the 4 + 4 positive helicity gravitini states of D = 4, N = 8 supergravity, (2.26) Applying these to (2.25) we obtain which ensures the correct action of the supersymmetry operator 4 with non-trivial equal time (anti)commutation relations: [cd] , and similarly for the remaining Q's. One can check this construction gives the correct action on the rest of the N = 8 multiplet and generalises to any dimension and number of supercharges. Note that in higher dimensions, where the little group is larger than u(1), the tensor product of two super Yang-Mills states typically yields a direct sum of supergravity states; to pick a specific component we need to project out the desired representation. To find the action of Q L ⊗ Q R on a state, we first act on the tensor product which contains it, and then project out the state we want. Returning to our maximal D = 5 example (described in (2.4) and table 2), we see that the gravitini states live in the (4; 1, 4) + (4; 4, 1) representation of so(3) ST ⊕ sp(2) ⊕ sp (2). Focusing on the (4; 1, 4) states, we see that they are obtained by a projection of A µ ⊗λ = (3; 1) ⊗ (2; 4) = (4; 1, 4) + (2; 1, 4). Then action of Q L and Q R , both living in (2; 4) of so(3) ST ⊕ sp(2), gives (4; 4, 1) + (2; 4, 1) + (2; 4, 5). Projecting out the gravitini, we find that the (4; 1, 4) states have been rotated into (4; 4, 1) states.

G algebras
The non-compact U-duality algebras of the supergravity theories appearing in the pyramid, figure 2, can be built straightforwardly using the tensor product of the left and right super Yang-Mills multiplets. Recall, the scalars of supergravity coupled to matter generated by squaring parametrise a G/H coset and T p (G/H) ∼ = p = g h, and so carry the p-representation of H. Consequently, the non-compact generators p, in a manifest int(N L , D) ⊕ int(N R , D) basis, can be read off from those tensor products which yield scalars, which are schematically given by: .

(2.32)
Acting twice on the vacuum, defined as the positive grade so(D − 4) singlet of the vector, yields the scalar states and the D − 4 vector states with zero so(2) charge living in, This contributes to p a term given by . λ a ⊗λ a : the scalars originating from λ L ⊗ λ R contribute a term given by 3. φ i ⊗φ i : the scalars originating from φ L ⊗ φ R contribute a term given by Bringing these elements together, we conclude that in total g as a vector space is given by: (2.41) In eq. (6) we present a set of commutators which define a Lie algebra structure on (2.41), giving precisely the algebras of the generalised U-duality pyramid in figure 2. To describe the complete set of commutators we use the left/right form of h ⊂ g given in (2.21), Table 6. Commutators of g(N L + N R , D)-algebra given in (2.41).
where we have distinguished the compact D[N L , N R ] c ⊂ h and the non-compact The non-trivial commutators amongst the compact generators have been given in (2.14). The remaining non-trivial commutators are given in eq. (6). Through out matrix multiplication is used except for where we have introduced the natural extension of the projector ϕ * , Note, for the sake of brevity we have reincorporated the D = 4, u(1) factor back into X L and X R , which therefore have equal and opposite traces. Moreover, leaving aside D = 3

JHEP01(2016)148
for the moment, the only non-vanishing α ⊕ β ∈ p L ⊕ p R occur in D = 4, the u(1) factors of N = 2, 1. See table 5. Simply regarding X L/R as tracefull generators belonging to a(N L/R ) automatically accounts for their action.
In three dimensions the formula can be simplified by "dualising" the A L ⊗ A R contributions into φ L ⊗ φ R terms. We no longer have the R L ⊗ R R term from tensoring the gauge fields, it is combined into a second R[N L , N R ] nc factor resulting the simplified D = 3 formula, together with a simplified set of commutation relations [39]. We conclude this discussion by relating this perspective back to the A = R, C, H, O framework developed in our previous work [31,33,39]. Each super Yang-Mills theory comes from reducing the fundamental D = 3, 4, 6, 10, N = 1 multiplet, and hence can be thought of in terms of those theories. This naturally associates R, C, H, O with each super Yang-Mills theory, according as to whether it came from the D = 3, 4, 6, 10 theory, respectivily. Equivalently, if one does not want to talk in terms of dimensional reduction, the associated division algebra is just A n with n = Q/2, where Q is the number of real supercharge components. The fermions can then be arranged into a single A n element.
where L x : A n → A n denotes left-multiplication by x.

Discussion
We have shown that the U-duality algebras g for all supergravity multiplets obtained by tensoring two super Yang-Mills multiplets in D ≥ 3 can be written in a single formula with three arguments, g(N L + N R , D). The formula relies on the link between the three JHEP01(2016)148 associative normed division algebras, R, C, H, and the representation theory of classical Lie algebras. The formula is symmetric under the interchange of N L and N R and provides another "matrix model", in the sense of Barton and Sudbery [32], for the exceptional Lie algebras. In this language the compact subalgebra h(N L + N R , D) has a simple form which makes the lef t ⊗ right structure clear. The non-compact p = g − h generators are obtained directly by examining the division algebraic representations carried by those left/right states that produce the scalar fields of the corresponding supergravity multiplets. Note, we are therefore implicitly assuming that the tensor product always gives supergravities with scalars parametrising a symmetric coset space. The only possible exception to this rule is given by N L = N R = 1. When there is a possible ambiguity in the coupling of the scalars it is resolved by the structure of the left and right symmetry algebras. For example, in D = 4 the N L = N R = 1 scalar coset manifold, , is the unique possibility consistent with the left and right super Yang-Mills data. This procedure gives all supergravity algebras with more than half-maximal supersymmetry. These cannot couple to matter, as reflected by the squaring procedure where only the fields of the supergravity multiplet are produced. However, for half-maximal and below, one can couple the theory to matter multiplets (vector or hyper). This does indeed happen when one squares; the fields obtained arrange themselves in the correct number of vector or hypermultiplets such that we fill up the entries of the generalised pyramid.
Theories with more general matter content do not naturally live in our pyramid, mainly because they lack an obvious division algebraic description. For example, the ST U model [93] is given by N = 2 supergravity in four dimensions coupled to three vector multiplets, while the entry for N = 2 in our pyramid necessarily comes coupled to a single hypermultiplet. Can squaring accommodate more general matter couplings? All factorized orbifold projections (as defined in [24]) of N = 8 supergravity can be obtained from the tensor product of the corresponding left and right orbifold projections of N = 4 super Yang-Mills multiplets [24]. This includes a large, but still restricted, class of matter coupled supergravities with specific U-dualities.
Theories coupled to an arbitrary number of vector multiplets can be obtained by tensoring a supersymmetric multiplet with a conveniently chosen collection of bosonic fields. In particular, here we consider an N R = 0 multiplet with a single gauge potential and n V scalar fields. The symmetries of the resulting supergravity multiplet are determined by the global symmetries postulated for the N R = 0 multiplet. We consider the simplest case where the n V scalar fields transform in the vector representation of a global SO(n V ). Following the procedure used to construct the generalised pyramid this uniquely fixes the global symmetries of the resulting supergravity multiplet and therefore, implicitly, the structure of the matter couplings. This idea is developed in the following section. We summarise the results 5 in table 7. Note, more generally these examples of factorisable N ≤ 4 matter coupled supergravities are also physically interesting. In particular, they JHEP01(2016)148 can be used to shed light on the UV divergences appearing in N ≤ 4 supergravity theories. Indeed, this double-copy construction of additional N = 4 vector multiplets was used in [18] to isolate the effect of the Marcus anomaly on N = 4 supergravity scattering amplitudes. Moreover, in [20] the dependence on n V was used to relate this anomaly to the 4-loop divergences appearing in these theories. 6

[N L ] V × [N R = 0] tensor products
Note that the general form for the maximally compact subgroups in the cosets given in table 7 is This is just the form,

3)
supergravity coupled to only vector multiplets by squaring since one always obtains at least one chiral multiplet when tensoring N = 1 SYM with a non-supersymmetric multiplet. The same applies to N = (1, 0) supergravity in 6 dimensions. These theories are interesting in their own right and will be analysed in forthcoming work [94]. 6 We thank one of our referees for bringing these developments to our attention.

JHEP01(2016)148
appearing in the generalised pyramid formula (2.21) with D[N L , N R = 0] = ∅. This is entirely consistent with the logic of the construction; we previously identified D[N L , N R ] with the generators Q L ⊗ Q R , which are clearly absent when N R = 0. The non-compact generators are also determined following the logic of the generalised pyramid presented in section 2.4, but now with only two scalar terms: A µ ⊗Ã ν and φ i ⊗φ i , whereφ i are the n V scalars transforming as a vector of SO(n V ).
The similarity transformation S n is trivially a bijective matrix algebra homomorphism and therefore also a Lie algebra isomorphism. Similarly, τ n is trivially an algebra isomorphism since τ itself is an algebra isomorphism. Therefore its restriction to ups(2n) is a Lie algebra isomorphism as the commutators are given by matrix commutators. Hence, θ n : usp(2n) → sa(n, H) (A.10) is a Lie algebra isomorphism, such that θ n (X † ) = θ n (X) † , θ n (X * ) = θ n (X) σ , θ n (X T ) = (θ n (X) † ) σ . (A.11) As an example, consider the most relevant case of n = 2: x z −z * y ∈ sa(2, H). holds for [X 1 , X 2 ] the usual matrix commutator, since (X 1 X 2 ) * = X 1 * X 2 * for X 1 , X 2 ∈ sa(n, A) (this is of course trivially true for A = R, C).