Exceptional groups, symmetric spaces and applications

Abstract

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.

40 pages, 1 figure

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• Lie Group Cosmology
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• On Multifield Born and Born-Infeld Theories and their non-Abelian Generalizations
• Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories
• On the Mathieu Conjecture for $Sp(N)$ and $G_2$
• Holographic RG flows in N=4 SCFTs from half-maximal gauged supergravity