Mapping the geometry of the E6 group

In this paper we present a construction for the compact form of the exceptional Lie group E6 by exponentiating the corresponding Lie algebra e6, which we realize as the the sum of f4, the derivations of the exceptional Jordan algebra J3 of dimension 3 with octonionic entries, and the right multiplication by the elements of J3 with vanishing trace. Our parametrization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E6 via a F4 subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F4. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E6 group manifold.Comment: 30 page

Symmetries of an Extended Hubbard Model

An extended Hubbard model with phonons is considered on a D-dimensional lattice. The symmetries of the model are studied in various cases. It is shown that for a certain choice of the parameters a superconducting SU_q(2) holds as a true quantum symmetry - but only for D=1. In a natural basis the symmetry requires vanishing local phonon coupling; a quantum symmetric Hubbard model without phonons can then be obtained by a mean field approximation.Comment: plain tex, 7 page

Squaring the Magic

We construct and classify all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras. Besides the known Freudenthal-Rozenfeld-Tits MS, the single-split G\"unaydin-Sierra-Townsend MS, and the double-split Barton-Sudbery MS, we obtain other 7 Euclidean and 10 Lorentzian novel MS's. We elucidate the role and the meaning of the various non-compact real forms of Lie algebras, entering the MS's as symmetries of theories of Einstein-Maxwell gravity coupled to non-linear sigma models of scalar fields, possibly endowed with local supersymmetry, in D = 3, 4 and 5 space-time dimensions. In particular, such symmetries can be recognized as the U-dualities or the stabilizers of scalar manifolds within space-time with standard Lorentzian signature or with other, more exotic signatures, also relevant to suitable compactifications of the so-called M*- and M'- theories. Symmetries pertaining to some attractor U-orbits of magic supergravities in Lorentzian space-time also arise in this framework.Comment: 21 pages, 1 figure, 20 tables; reference adde

Mapping the geometry of the F4 group

In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2=F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.Comment: 50 pages; some typos correcte

Duality, Entropy and ADM Mass in Supergravity

We consider the Bekenstein-Hawking entropy-area formula in four dimensional extended ungauged supergravity and its electric-magnetic duality property. Symmetries of both "large" and "small" extremal black holes are considered, as well as the ADM mass formula for N=4 and N=8 supergravity, preserving different fraction of supersymmetry. The interplay between BPS conditions and duality properties is an important aspect of this investigation.Comment: 45 pages, typos corrected, references adde

Exceptional groups, symmetric spaces and applications

In this article we provide a detailed description of a technique to obtain a simple parametrization for different exceptional Lie groups, such as G2, F4 and E6, based on their fibration structure. For the compact case, we construct a realization which is a generalization of the Euler angles for SU(2), while for the non compact version of G2(2)/SO(4) we compute the Iwasawa decomposition. This allows us to obtain not only an explicit expression for the Haar measure on the group manifold, but also for the cosets G2/SO(4), G2/SU(3), F4/Spin(9), E6/F4 and G2(2)/SO(4) that we used to find the concrete realization of the general element of the group. Moreover, as a by-product, in the simplest case of G2/SO(4), we have been able to compute an Einstein metric and the vielbein. The relevance of these results in physics is discussed.Comment: 40 pages, 1 figur

Euler angles for G2

We provide a simple parametrization for the group G2, which is analogous to the Euler parametrization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.Comment: 21 pages, 2 figures, some misprints correcte

Iwasawa N=8 Attractors

Starting from the symplectic construction of the Lie algebra e_7(7) due to Adams, we consider an Iwasawa parametrization of the coset E_7(7)/SU(8), which is the scalar manifold of N=8, d=4 supergravity. Our approach, and the manifest off-shell symmetry of the resulting symplectic frame, is determined by a non-compact Cartan subalgebra of the maximal subgroup SL(8,R) of E_7(7). In absence of gauging, we utilize the explicit expression of the Lie algebra to study the origin of E_7(7)/SU(8) as scalar configuration of a 1/8-BPS extremal black hole attractor. In such a framework, we highlight the action of a U(1) symmetry spanning the dyonic 1/8-BPS attractors. Within a suitable supersymmetry truncation allowing for the embedding of the Reissner-Nordstrom black hole, this U(1) is interpreted as nothing but the global R-symmetry of pure N=2 supergravity. Moreover, we find that the above mentioned U(1) symmetry is broken down to a discrete subgroup Z_4, implying that all 1/8-BPS Iwasawa attractors are non-dyonic near the origin of the scalar manifold. We can trace this phenomenon back to the fact that the Cartan subalgebra of SL(8,R) used in our construction endows the symplectic frame with a manifest off-shell covariance which is smaller than SL(8,R) itself. Thus, the consistence of the Adams-Iwasawa symplectic basis with the action of the U(1) symmetry gives rise to the observed Z_4 residual non-dyonic symmetry.Comment: 1+26 page