Seven sphere and the exceptional nonlinear superconformal algebras

The 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 are reviewed. Both the N = 8 and N = 7 algebras admit unitary realizations 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. They can also be realized over the coset spaces SO(8) × U(1)/SO(7) and SO(7) × U(1)/G₂ for some fixed values of their central charges, respectively. The coset space SO(8)/SO(7) is the seven-sphere S⁷, whereas the space SO(7)/G₂ represents the seven-sphere with torsion. We conclude with a discussion of a novel '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.