Abstract
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).
Highlights
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
The D = 6 square as the Freudenthal-Rosenfeld-Tits magic square restricted to the subgroups that commute with a single quaternionic structure
We began with the observation developed in [41] that N = 2m-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 AnN
Summary
An algebra A defined over R with identity element e0, is said to be composition if it has a non-degenerate quadratic form n : A → R such that, n(ab) = n(a)n(b), ∀ a, b ∈ A,. We can define an order three Lie algebra automorphism θ : tri(A) → tri(A) : (A, B, C) → (B, C, A),. A non-compact real form gnc of a complex semi-simple Lie algebra gC admits a symmetric decomposition gnc = h + p,. If a compact real form gc shares with some non-compact real form gnc a common subalgebra, gnc = h + p and gc = 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, h is the maximal compact subalgebra of gnc This observation is sufficient to confirm that our construction yields the real forms in table 1 and we can identify.
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