A unified approach to the minimal unitary realizations of noncompact groups and supergroups

Abstract

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 × SL(2, R) such that G/H × 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 unitary representations 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 × 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 even subgroups SO(n) × SL(2, R) are the singleton representations. We also give the minimal realization of the one parameter family of Lie superalgebras D (2, 1; σ).

Exceptional F (4) higher-spin theory in AdS6 at one-loop and other tests of duality

“…Furthermore there exists a one-to-one correspondence between the minimal unitary representations of SO(d, 2) and their deformations and massless conformal fields in d dimensional Minkowskian spacetimes [8]. These results were obtained by quantization of the geometric realization of SO(d, 2) as quasiconformal groups [38][39][40]. The geometric realizations of noncompact groups as quasiconformal groups that leave invariant a quartic light-cone was discovered in [38].…”

Minimal unitary representation of 5d superconformal algebra F(4) and AdS6Fernando1, Günaydın2 2015Nuclear Physics BSelf Cite

“…The quasiconformal realization of the minimal unitary representation of symplectic groups coincides with the realization as bilinear of oscillators [16] and the Joseph ideal vanishes identically [9]. That is why the enveloping algebra of the singletonic realization of Sp(4, R) ≈ SO (3,2) as bilinears of covariant twistorial oscillators leads directly to the Fradkin-Vasiliev higher spin algebra in AdS 4 .…”

“…We shall first review the construction of the minrep of SO(5, 2) by quantization of its geometric realization as a quasiconformal group following [16]. We show that it corresponds to a conformal massless scalar field in five dimensions.…”

Relativistic harmonic oscillator revisited

“…However, this structure of double creators or double annihilators is tailor made for the oscillator approach to representation theory for non-compact groups or supergroups developed in [21]- [24]. Using those techniques we will classify the states as parts of infinite dimensional unitary representations as explained below.…”

“…This type of oscillator representation was not previously considered in [21]- [24]. The new non-linear expressions for the generators given in Eqs.…”