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Register a new userRealizations of exceptional U-duality groups as conformal and quasiconformal groups and their minimal unitary representations
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M. Gunaydin
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We review the novel quasiconformal realizations of exceptional U-duality groups whose quantization lead directly to their minimal unitary irreducible representations. 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 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. These conformal and quasiconformal groups act as spectrum generating symmetry groups in five and four dimensions and are isomorphic to the U-duality groups of the corresponding supergravity theories in four and three dimensions, respectively. Hence the spectra of these theories are expected to form unitary representations of these groups whose minimal unitary realizations are also reviewed.
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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.
We present a nonlinear realization of E_8 on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined ``light cone in 57 dimensions. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E_7 on a 27 dimensional vector space which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned.
We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra dilatonic coordinate, whose rotation, Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra cocycle coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.
Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the mass parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N+1)-th order equation of motion. As a special case, the Schroedinger group and the standard Newton equations are obtained for N=1 (l=1/2).
We study the minimal unitary representation (minrep) of SO(4,2) over an Hilbert space of functions of three variables, obtained by quantizing its quasiconformal action on a five dimensional space. The minrep of SO(4,2), which coincides with the minrep of SU(2,2) similarly constructed, corresponds to a massless conformal scalar in four spacetime dimensions. There exists a one-parameter family of deformations of the minrep of SU(2,2). For positive (negative) integer values of the deformation parameter zeta one obtains positive energy unitary irreducible representations corresponding to massless conformal fields transforming in (0,zeta/2) ((-zeta/2,0)) representation of the SL(2,C) subgroup. We construct the supersymmetric extensions of the minrep of SU(2,2) and its deformations to those of SU(2,2|N). The minimal unitary supermultiplet of SU(2,2|4), in the undeformed case, simply corresponds to the massless N=4 Yang-Mills supermultiplet in four dimensions. For each given non-zero integer value of zeta, one obtains a unique supermultiplet of massless conformal fields of higher spin. For SU(2,2|4) these supermultiplets are simply the doubleton supermultiplets studied in arXiv:hep-th/9806042.
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