### Abstract

The realizations of the exceptional non-linear (quadratically generated, 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 free fermions in 8 of Spin(7) and 7 of G_{2}, with the central charges c_{8} = 26/5 and c_{7} = 5, respectively. They can also be realized over the coset spaces SO(8) × U(1)/SO(7) and SO(7) × 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 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.

Original language | English (US) |
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Pages (from-to) | 70-75 |

Number of pages | 6 |

Journal | Nuclear Physics B - Proceedings Supplements |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1997 |

### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics
- Nuclear and High Energy Physics