### Abstract

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N = 2 Maxwell–Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N = 2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N = 2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.

Original language | English (US) |
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Title of host publication | The Attractor Mechanism - Proceedings of the INFN-Laboratori Nazionali di Frascati School 2007 |

Editors | Stefano Bellucci |

Publisher | Springer Science and Business Media, LLC |

Pages | 31-84 |

Number of pages | 54 |

ISBN (Print) | 9783642107351 |

DOIs | |

State | Published - Jan 1 2010 |

Event | 3rd School on Attractor Mechanisms, SAM 2007 - Frascati, Italy Duration: Jun 18 2007 → Jun 22 2007 |

### Publication series

Name | Springer Proceedings in Physics |
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Volume | 134 |

ISSN (Print) | 0930-8989 |

ISSN (Electronic) | 1867-4941 |

### Other

Other | 3rd School on Attractor Mechanisms, SAM 2007 |
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Country | Italy |

City | Frascati |

Period | 6/18/07 → 6/22/07 |

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### All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

### Cite this

*The Attractor Mechanism - Proceedings of the INFN-Laboratori Nazionali di Frascati School 2007*(pp. 31-84). (Springer Proceedings in Physics; Vol. 134). Springer Science and Business Media, LLC. https://doi.org/10.1007/978-3-642-10736-8_2