Seven-sphere and the exceptional N = 7 and N = 8 superconformal algebras

We study realisations 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 highest-weight 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 S⁷, whereas the space SO(7)/G₂ represents the seven-sphere with torsion, S⁷_T. The division algebra 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.