Seven–sphere and the Exceptional N=7 and N=8 Superconformal Algebras

We study realizations of the exceptional non-linear (quadratically generated, or W type) N = 8 and N = 7 superconformal algebras with Spin(7) and G2 affine symmetry currents, respectively. Both the N = 8 and N = 7 algebras admit unitary highestweight representations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G2, with the central charges c8 = 26/5 and c7 = 5, respectively. Furthermore, we show that the general coset Ansätze for the N = 8 and N = 7 algebras naturally lead to the coset spaces SO(8) × U(1)/SO(7) and SO(7) × U(1)/G2 , respectively, as the additional consistent solutions for certain values of the central charge. The coset space SO(8)/SO(7) is the seven-sphere S, whereas the space SO(7)/G2 represents the sevensphere with torsion, S T. The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realizations. We also comment on some possible applications of our results to string theory. Supported in part by the ‘Deutsche Forschungsgemeinschaft’ and the ‘Volkswagen Stiftung’ On leave of absence from: High Current Electronics Institute of the Russian Academy of Sciences, Siberian Branch, Akademichesky 4, Tomsk 634055, Russia