arXiv:0901.1646v2 [hep-th] 2 Mar 2009 Preprint typeset in JHEP style - HYPER VERSION Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors Murat G¨unaydin†∗ †Institute for Gravitation and the Cosmos Physics Department Penn State University University Park, PA 16802, USA Oleksandr Pavlyk‡† ‡Wolfram Research Inc. 100 Trade Center Dr. Champaign, IL 61820, USA Abstract:After reviewing the algebraic structures that underlie the geometries ofN= 2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups.They areF4(4), E6(2), E7(−5), E8(−24)and SO(n+ 2,4).Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which areSL(3,R),SL(3,C),SU∗(6), E6(6)andSO(n−1,1)×SO(1,1), respectively. We determine the spherical vectors of quasi- conformal realizations of all these groups twisted by a unitary character. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic ground- work for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series.For rank 2 cases, SU(2,1) andG2(2), corresponding to simpleN= 2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries ofN≥4 supergravity theories. They lead to quasiconfor- mal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms. Keywords:Supergravity, Duality, Black Holes. ∗murat@phys.psu.edu †pavlyk@wolfram.com
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