Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

  • Murat GünaydinEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 134)


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.


Black Hole Black Hole Solution Jordan Algebra Supergravity Theory Twistor Space 
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I would like to thank Stefano Bellucci for his kind invitation to deliver these lectures at SAM 2007 and the participants for numerous stimulating discussions. I wish to thank Stefano Bellucci, Sergio Ferrara, Kilian Koepsell, Alessio Marrani, Andy Neitzke, Hermann Nicolai, Oleksandr Pavlyk, Boris Pioline and Andrew Waldron for enjoyable collaborations and stimulating discussions on various topics covered in these lectures. Thanks are also due to the organizers of the “Fundamental Aspects of Superstring Theory 2009” Workshop at KITP, UC Santa Barbara and of the New Perspectives in String Theory 2009 Workshop at GGI, Florence where part of these lectures were written up. This work was supported in part by the National Science Foundation under grant number PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.


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Authors and Affiliations

  1. 1.Institute for Gravitation and the Cosmos, Physics DepartmentPenn State UniversityUniversity ParkUSA

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