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Résumé |
We begin by reviewing the theme of "gravity as the double-copy of Yang-Mills theory”, particularly in the context of scattering amplitudes. This paradigm has witnessed some remarkable recent developments, perhaps most notably the conjecture, underpinned by Bern-Carrasco-Johansson colour/kinematic duality, that n-point graviton amplitudes are the double-copy of n-point gluon amplitudes to all orders. Relations of this kind raise the question: to what extent, or in what sense, can one really regard quantum gravity as the square of Yang-Mills theory? Motivated in part by such considerations we introduce a dictionary relating the fields of (super)gravity to those of two (super) Yang-Mills theories via a convolutive tensor product that makes use of a global bi-adjoint scalar field with cubic interactions. The first test such a dictionary must pass is at the level of symmetries. To linear approximation it is found that the (super)gravity symmetries of general covariance, p-form gauge, local Lorentz and local supersymmetry are generated by those of the two (super) Yang-Mills theories, namely local gauge and global (super) Poincaré symmetries. In the context of string/M-theory the global symmetries of supergravity, which are intimately related to the notion of U-duality, are of equal importance. These manifest themselves most clearly in D=3 spacetime dimensions and in this case our dictionary yields an intriguing result: the U-dualities generated by the product of two D=3, N=1,2,4,8 super Yang-Mills theories are given the Freudenthal–Rosenfeld–Tits magic square, a symmetric 4x4 array of Lie algebras. This surprise is neatly explained by the observation that the four D=3, N=1,2,4,8 super Yang-Mills theories can be uniformly articulated in terms of the four division algebra: the reals R, complexes C, quaternions H and octonions O. The magic square itself is constructed over the possible products of two division algebras, directly linking these physical and mathematical squares. Exploiting the relationship between minimal super Yang-Mills theories in D=3,4,6,10 and R,C,H,O this construction is extended to include all possible cases for D=3 to 10, yielding a magic pyramid of Lie algebras. We conclude with some more speculative implications of the dictionary, focussing on D=6, N=(2,0) superconformal theories. |
