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

Abstract

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,$\mathbb{R}$), SL(3,$\mathbb{C}$), 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 5D 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.