Unitary realizations of U-duality groups as conformal and quasiconformal groups and extremal black holes of supergravity theories

We review the current status of the construction of unitary representations of U-duality groups of supergravity theories in five, four and three dimensions. We focus mainly on the maximal N = 8 supergravity theories and on the N = 2 Maxwell-Einstein supergravity (MESGT) theories defined by Jordan algebras of degree three in five dimensions and their descendants in four and three dimensions. Entropies of the extremal black hole solutions of these theories in five and four dimensions are given by certain invariants of their U-duality groups. The five dimensional U-duality groups admit extensions to spectrum generating generalized conformal groups which are isomorphic to the U-duality groups of corresponding four dimensional theories. Similarly, the U-duality groups of four dimensional theories admit extensions to spectrum generating quasiconformal groups that are isomorphic to the corresponding U-duality groups in three dimensions. For example, the group E ₈₍₈₎ can be realized as a quasiconformal group in the 57 dimensional charge-entropy space of BPS black hole solutions of maximal N = 8 supergravity in four dimensions and leaves invariant "lightlike separations" with respect to a quartic norm. Similarly E ₇₍₇₎ acts as a generalized conformal group in the 27 dimensional charge space of BPS black hole solutions in five dimensional N = 8 supergravity and leaves invariant "lightlike separations" with respect to a cubic norm. For the exceptional N = 2 Maxwell-Einstein supergravity theory the corresponding quasiconformal and conformal groups are E _8(-24) and E _7(-25), respectively. We outline the oscillator construction of the unitary representations of generalized conformal groups that admit positive energy representations, which include the U-duality groups of N = 2 MESGT's in four dimensions. We conclude with a review of the minimal unitary realizations of U-duality groups that are obtained by quantizations of their quasiconformal actions and discuss in detail the minimal unitary realization of E ₈₍₈₎.