Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-04-10
JHEP 0609:050,2006
Physics
High Energy Physics
High Energy Physics - Theory
Minor revision to improve the presentation. Typos corrected. Version to appear in JHEP. 35 pages; Latex file
Scientific paper
10.1088/1126-6708/2006/09/050
We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones, defined by a quartic norm, invariant and have maximal rank subgroups of the form H X SL(2,R) such that G/H X SL(2,R) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A_2, G_2, D_4, F_4, E_6, E_7, E_8 and Sp(2n,R). The minimal UIRs of Sp(2n,R) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU(m,n), SO(m,n), SO*(2n) and SL(m,R) are also given explicitly. We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H X SL(2,R). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F(4) and OSp(n|2,R) (in the degenerate limit). The minimal unitary representations of OSp(n|2,R) with the even subgroup SO(n) X SL(2,R) are the singleton representations. We also give the minimal realization of one parameter family of Lie superalgebras D(2,1;\sigma).
Gunaydin Murat
Pavlyk Oleksandr
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