Minimal Unitary Realizations of Exceptional U-duality Groups and Their Subgroups as Quasiconformal Groups

We study the minimal unitary representations of noncompact exceptional groups that arise as U-duality groups in extended supergravity theories. First we give the unitary realizations of the exceptional group E8(−24) in SU*(8) as well as SU(6,2) covariant bases. E8(−24) has E7 × SU(2) as its maximal compact subgroup and is the U-duality group of the exceptional supergravity theory in d = 3. For the corresponding U-duality group E8(8) of the maximal supergravity theory the minimal realization was given in [1]. The minimal unitary realizations of all the lower rank noncompact exceptional groups can be obtained by truncation of those of E8(−24) and E8(8). By further truncation one can obtain the minimal unitary realizations of all the groups of the "Magic Triangle". We give explicitly the minimal unitary realizations of the exceptional subgroups of E8(−24) as well as other physically interesting subgroups. These minimal unitary realizations correspond, in general, to the quantization of their geometric actions as quasi-conformal groups as defined in [2].