Volume 130B. number 1,2 PHYSICS LETTERS 13 October 1983 SEVEN-DIMENSIONAL DE SITTER AND SIX-DIMENSIONAL CONFORMAL SUPERSYMMETRIES Z. HASIEWICZ Institute for Theorettcal Physics, Cybulsktego 36, 50-205 Wrocl'aw,Poland J. LUKIERSKI 1 Laboratotre de Physique Thdortque a, UmversttOde Bordeaux L Rue de Solartum, 331 70 Gradtgnan, France and P. MORAWIEC Instttute for Theoretwal Physics, Cybulsktego 36, 50-205 Wrocgaw,Poland Received 24 May 1983 The N-extended D = 7 De Sitter algebras UaU(4, N, H) with bosonic sector SO(6, 2) XSp(N), describing also N-extended D = 6 superconformal symmetries are given. The quatermonic structure of real representation is described by three real quaterniomc Majorana condltlon~ We conjecture thatN = 2D = 7 De Satter supergroup UaU(4,2; Iq) describes the super- symmetries of the vacuum solutaon for D = 11 supergravity, with the space-time described by seven-dimensional De Sitter space and internal space S 4. 1. Recently Salam and Sezgln [1] considered maxi- mally extended D = 7 supergravlty [with Sp(2) = USp(4; C) -~ SO(5) internal symmetiy] and Townsend and Van Nieuwenhmzen [2] considered slmple gauged D = 7 supergravlty [with Sp(1) = USp(2; C) -~ SO(3) internal symmetry]. Also recently several authors [3-5] considered a six-dimensional supersymmetnc formalism in order to understand in a geometric way via dimensional reduction N = 2 supersymmetry in four dimensions. In this paper we shall present ex- tended graded De Sitter algebras in D = 7, whach describe D = 6 superconformal supersymmetrles also. From the algebraic point of view the superalgebras in D = 4, 5 and 7 are distinguished because only in these dimensmns (for d ~> 4) N-extended graded De Sitter algebras wath a family of internal symmetries do exist, described respectively by three compact classi- cal groups" O(N) (d = 4), U(N) (d = 5) and Sp(N) = USp(2N, C)(d = 7). These three famdles of superal- 1 On leave of absence from the Institute for Theoretical Physics, Unwerslty of Wroct'aw, Poland. 2 Eqmpe de Recherche assoc16e au CNRS. gebras were firstly written down by Nahm [6] and due to the isomorphlsms Spin (3,2) = Sp(4, FI), Spin (4,2) = SU(2,2) = Ua(4, C) 4:1 and Spin (6,2) = SO*(8; C) = SO(4; H) = Ua(4; H) they look as follows" d = 4: OSp(N;4) (bosonlc sector Sp(4; tq) × 0(N)), (la) d = 5: SU(2,2,N) (bosonic sector SU(2,2) × U(N)), and ,2 (lb) 4:1 ~_2 We denote by Ua(2n; F) the antiunitary n × n matrix group with F-valued (F = FI, C, H) entnes, preserving the antl-hermltean bilinear form (see e.g. ref. [7]). We have Ua(2n, FI) = Sp(2n, Iq), Ua(2n, C) = U(n, n) and Ua(2n, H) = O(2n, H) = O*(4n, C). The quaternionic supergroup U~U(4;N; H) due to the relations Ua(4) = 0(4, H) = U(4,4) n 0(8; C) and U(N; H) = Sp(N) = U(2N) n Sp(2n; C) can be described as SU(4,4;2N) n Sp(8, 2n; 12). The quatermonic super- groups preserving bflinear quaternion-valued invanants were mentaoned by Kac [8] and discussed further m refs. [9-111. 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 55