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

TL;DR

This work develops minimal unitary representations for noncompact exceptional U‑duality groups arising in extended supergravity, focusing on $E_{8(-24)}$ and its relation to $E_{8(8)}$. It constructs explicit minimal realizations in $SU^{*}(8)$ and $SU(6,2)$ bases, employing a 5‑grading and a geometric, quasi‑conformal action on a 57‑dimensional space, with a corresponding quartic invariant $I_4$ driving the realization. The authors then implement a web of consistent truncations to lower‑rank, quasi‑conformal subalgebras ($E_{7(-5)}$, $E_{6(2)}$, $E_{6(-14)}$, $F_{4(4)}$, $SO(4,4)$, and further to $SO(2p,2)$). These truncations preserve the minimal unitary structure and yield c‑number Casimirs, illustrating how the same framework extends to the subgroups and to the subalgebra chain terminating in the magic triangle. Overall, the paper provides a cohesive, quantized realization of exceptional U‑duality groups as quasiconformal actions, with implications for spectrum‑generating symmetries in M‑theory and related supergravity theories.

Abstract

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 E_{8(-24)} in SU*(8) as well as SU(6,2) covariant bases. E_{8(-24)} has E_7 X 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 E_{8(8)} of the maximal supergravity theory the minimal realization was given in hep-th/0109005. The minimal unitary realizations of all the lower rank noncompact exceptional groups can be obtained by truncation of those of E_{8(-24)} and E_{8(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 E_{8(-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 hep-th/0008063.