Global symmetries of Yang-Mills squared in various dimensions

Published for SISSA by Springer Received: November 17, 2015 Accepted: January 3, 2016 Published: January 25, 2016 A. Anastasiou,a L. Borsten,a,b M.J. Hughesa and S. Nagya,c a Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, U.K. b School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland c Department of Mathematics, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal E-mail: , , , Abstract: 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 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. Keywords: Supersymmetry and Duality, Extended Supersymmetry, Supergravity Models, Global Symmetries ArXiv ePrint: 1502.05359 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP01(2016)148 JHEP01(2016)148 Global symmetries of Yang-Mills squared in various dimensions Contents 1 2 Global symmetries of super Yang-Mills squared 2.1 Tensoring super Yang-Mills theories in D ≥ 3 2.2 R-symmetry algebras 2.3 H algebras 2.4 G algebras 4 4 5 10 14 3 Discussion 3.1 [NL ]V × [NR = 0] tensor products 18 20 A Scalars in D[N ] 22 1 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 [2–27]. Indeed, invoking the BernCarrasco-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], L3 (ANL , ANR ) := tri(ANL ) ⊕ tri(ANR ) + 3(ANL ⊗ ANR ), –1– (1.1) JHEP01(2016)148 1 Introduction ANL \ANR C H O R N = 2, f = 4 G = SL(2, R) H = SO(2) N = 3, f = 8 G = SU(2, 1) H = SO(3) × SO(2) N = 5, f = 16 G = USp(4, 2) H = SO(5) × SO(3) N = 9, f = 32 G = F4(−20) H = SO(9) C N = 3, f = 8 G = SU(2, 1) H = SO(3) × SO(2) N = 4, f = 16 G = SU(2, 1)2 H = SO(3)2 × SO(2)2 N = 6, f = 32 G = SU(4, 2) H = SO(6) × SO(3) × SO(2) N = 10, f = 64 G = E6(−14) H = SO(10) × SO(2) H N = 5, f = 16 G = USp(4, 2) H = SO(5) × SO(3) N = 6, f = 32 G = SU(4, 2) H = SO(6) × SO(3) × SO(2) N = 8, f = 64 G = SO(8, 4) H = SO(8) × SO(3) × SO(3) N = 12, f = 128 G = E7(−5) H = SO(12) × SO(3) O N = 9, f = 32 G = F4(−20) H = SO(9) N = 10, f = 64 G = E6(−14) H = SO(10) × SO(2) N = 12, f = 128 G = E7(−5) H = SO(12) × SO(3) N = 16, f = 256 G = E8(8) H = SO(16) Table 1. (N = NL + NR )-extended D = 3 supergravities obtained by left/right super Yang-Mills multiplets with NL , NR = 1, 2, 4, 8. The algebras of the corresponding U-duality groups G and their maximal compact subgroups H are given by the magic square of Freudenthal-Rosenfeld-Tits [34– 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 ANL , ANR = R, C, H, O, where we have adopted the convention that dim AN = N . The triality algebra of A, denoted tri(A), is related to the total on-shell global symmetries of the associated super YangMills 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 AN = R, C, H, O description of D = 3, N = 1, 2, 4, 8 super Yang-Mills theories. 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 [40–44]. 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 [41–85] 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, 20 , 21 , . . . 27 supercharges.2 The subset of algebras in the trapezoid given by D = 3, 4, 5 and 25 , 26 , 27 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. + + + It also includes the affine Kac-Moody algebras, e9 = e+ 8 , e7 , e6 , so10 in D = 2 as made more precise in [86]. 2 –2– JHEP01(2016)148 R 1. The algebra ra(NL + NR , D) of (NL + NR )-extended R-symmetry in D dimensions, ra(NL + NR , D) = a(NL , D) ⊕ a(NR , D) + D[NL , NR ]; (1.2) 2. The algebra h(NL + NR , D) of H, the maximal compact subgroup of the U-duality group G, h(NL + NR , D) = int(NL , D) ⊕ int(NR , D) ⊕ δD,4 u(1) + D[NL , NR ]; (1.3) 3. The algebra g(NL + NR , D) of the U-duality group (...truncated)