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arXiv · 0904.0784

Quasiconformal Realizations of E_{6(6)}, E_{7(7)}, E_{8(8)} and SO(n+3,m+3), N=4 and N>4 Supergravity and Spherical Vectors

Murat Gunaydin·Oleksandr Pavlyk
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Abstract

After reviewing the underlying algebraic structures we give a unified realization of split exceptional groups F_{4(4)},E_{6(6)}, E_{7(7)}, E_{8(8)} and of SO(n+3,m+3) as quasiconformal groups that is covariant with respect to their (Lorentz) subgroups SL(3,R), SL(3,R)XSL(3,R), SL(6,R), E_{6(6)} and SO(n,m)XSO(1,1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character $ν$. We also give their quadratic Casimir operators and determine their values in terms of $ν$ and the dimension $n_V$ of the underlying Jordan algebras. For $ν= -(n_V+2)+iρ$ the quasiconformal action induces unitary representations on the space of square integrable functions in $(2n_V+3)$ variables, that belong to the principle series. For special discrete values of $ν$ the quasiconformal action leads to unitary representations belonging to the discrete series and their continuations. The manifolds that correspond to "quasiconformal compactifications" of the respective $(2n_V+3)$ dimensional spaces are also given. We discuss the relevance of our results to N=8 supergravity and to N=4 Maxwell-Einstein supergravity theories and, in particular, to the proposal that three and four dimensional U-duality groups act as spectrum generating quasiconformal and conformal groups of the corresponding four and five dimensional supergravity theories, respectively.

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Murat Gunaydin, Oleksandr Pavlyk. 2009-04-06. Quasiconformal Realizations of E_{6(6)}, E_{7(7)}, E_{8(8)} and SO(n+3,m+3), N=4 and N>4 Supergravity and Spherical Vectors. https://arxiv.org/abs/0904.0784

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