A magic pyramid of supergravities
A. Anastasiou
L. Borsten
M.J. Duff
L.J. Hughes
S. Nagy
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-RosenfeldTits 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 4 4 magic square, while the higher levels are comprised of 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 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) Sp(1).
1 Introduction 2 3 4
The magic square
The magic pyramid
3.1 Super Yang-Mills over R, C, H, O
3.2 The D = 3 magic square of supergravities
3.3 Generalisation to D = 4, 6, 10
3.4 Complex and quaternionic structures
3.5 S-duality of N = 4 SYM and supergravity
The conformal magic pyramid
4.1 Barton-Sudbery-style formula Conclusions A D = 6 tensoring tables 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 [610]. 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
Mtheory 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], NL and NR super Yang-Mills
F4(20)
E6(14)
E7(5)
F4(20)
E6(14)
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 (An, AnNL , AnNR ),
one for spacetime and two for the left/right Yang-Mills multiplets.
In previous work [15] we built a symmetric 44 array of three-dimensional supergravity
multiplets, with N = NL + NR, by tensoring a left NL = 1, 2, 4, 8 SYM multiplet with
a right NR = 1, 2, 4, 8 SYM multiplet. Remarkably, the corresponding U-dualities filled
out the Freudenthal-Rosenfeld-Tits magic square [1621]; a symmetric 4 4 array of Lie
algebras defined by a single formula taking as its argument a pair of division algebras,
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 square1 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 ANL -valued
super Yang-Mills multiplet with an ANL -valued super Yang-Mills multiplet yields a
supergravity (...truncated)