Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors

• Published in: The journal of high energy physics Vol. 2010; no. 4
• Main Authors: Günaydin, Murat, Pavlyk, Oleksandr
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2010; Springer Nature B.V
• Subjects: Algebra; Classical and Quantum Gravitation; Elementary Particles; Fields (mathematics); Group theory; High energy physics; Mathematical analysis; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; Representations; String Theory; Supergravity; Symmetry; Vectors (mathematics); Black Holes in String Theory; Supergravity Models; Supersymmetry and Duality; Extended Supersymmetry
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP04(2010)070

After reviewing the algebraic structures that underlie the geometries of N = 2 Maxwell-Einstein supergravity theories (MESGT) with symmetric scalar manifolds in five and four dimensions, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F 4(4) , E 6(2) , E 7(−5 ), E 8(−24) and SO( n +2, 4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3, ), SL(3, ), SU *(6), E 6(−26) and SO( n − 1, 1) × SO(1, 1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character ν . We present their quadratic Casimir operators and determine their values in terms of ν and the number n V of vector fields of the respective 5 D supergravity. For ν = −( n V + 2) + iρ the quasiconformal action induces unitary representations belonging to the principal series. For special discrete values of ν it leads to unitary representations belonging to the quaternionic discrete series. Our results lay the algebraic groundwork for constructing explicitly the quaternionic discrete series unitary representations. For rank 2 cases, SU(2, 1) and G 2(2) , corresponding to simple N = 2 supergravity in four and five dimensions, respectively, this program was carried out in arXiv:0707.1669 and applied to quantum attractor flows.