7-SPHERE AND THE EXCEPTIONAL NONLINEAR SUPERCONFORMAL ALGEBRAS

The realizations of the exceptional non-linear (quadratically generate d, or W-type) N = 8 and N = 7 superconformal algebras with Spin(7) and G(2) affine symmetry currents are reviewed. Both the N = 8 and N = 7 algebras admit unitary realizations in terms of a single boson and fre e fermions in 8 of Spin(7) and 2 of G(2), with the central charges c(8 ) = 26/5 and c(7) = 5, respectively. They can also be realized over th e coset spaces SO(8) x U(1)/SO(7) and SO(7) x U(1)/G(2) for some fixed values of their central charges, respectively. The coset space SO(8)/ SO(7) is the seven-sphere S-7, whereas the space SO(7)/G(2) represents the seven-sphere with torsion. We conclude with a discussion of a nov el 'hybrid' method developed recently that yields unitary realizations of the exceptional N = 8 and N = 7 algebras for all allowed values of their central charges.