7-SPHERE AND THE EXCEPTIONAL N=7 AND N=8 SUPERCONFORMAL ALGEBRAS

We study realisations of the exceptional non-linear (quadratically gen erated, or W-type) N = 8 and N = 7 superconformal algebras with Spin(7 ) and G(2) affine symmetry currents, respectively. Both the N = 8 and N = 7 algebras admit unitary highest-weight representations in terms o f a single boson and free fermions in (8) under bar of Spin(7) and (7) under bar of G(2), With the central charges c(8) = 26/5 and c(7) = 5, respectively. Furthermore, we show that the general coset ansatze for the N = 8 and N = 7 algebras naturally lead to the coset spaces SO(8) x U(1)/SO(7) and SO(7) x U(1)/G(2), respectively, as the additional c onsistent solutions for certain values of the central charge. The cose t space SO(8)/SO(7) is the seven-sphere S-7, whereas the space SO(7)/G (2) represents the seven-sphere with torsion, S-T(7). The division alg ebra of octonions and the associated triality properties of SO(8) play an essential role in all these realisations, We also comment on some possible applications of our results to string theory.