Abstract
We study the Hamiltonian dynamics of a five-dimensional Chern–Simons theory for the gauge algebra C_5 of Izaurieta, Rodriguez and Salgado, the so-called hbox {S}_H-expansão of the 5D (anti-)de Sitter algebra (a)ds, based on the cyclic group {mathbb {Z}}_4. The theory consists of a 1-form field containing the (a)ds gravitation variables and 1-form field transforming in the adjoint representation of (a)ds. The gravitational part of the action necessarily contains a term quadratic in the curvature, beyond the Einstein–Hilbert and cosmological terms, for any choice of the two independent coupling constants. The total action is also invariant under a new local symmetry, called “crossed diffeomorphisms”, beyond the usual space-time diffeomorphisms. The number of physical degrees of freedom is computed. The theory is shown to be “generic” in the sense of Bañados, Garay and Henneaux, i.e., the constraint associated to the time diffeomorphisms is not independent from the other constraints.
Highlights
An analysis of the phenomenological aspects of 5D Chern–Simons gravity models with dimensional reduction to 4D have been investigated, with results indicating their relevance as physical theories [8,9,10,11,12,13]
Works by Bañados, Garay and Henneaux [16,17] have made significant advances, showing the existence of so-called “generic” theories, where the constraint associated with the time diffeomorphisms is no longer independent, but can be seen as a combination of the constraints associated with gauge invariance and spatial diffeomorphisms
First, that the theory depends on two independent coupling constants,2 second, that it is generic in the sense defined above, and third, that it is invariant under a new class of diffeomorphisms specific to these expanded algebra, which we call “crossed diffeomorphisms”
Summary
We will present some results known in the literature on the Chern–Simons gravitation theories, important for the understanding of this work. Chern–Simons theories occur only in odd dimensions. To construct the Chern–Simons action corresponding to (A)dS5, we use the formalism of the differential forms. The fields are given by the 1-form connection A = Aμd xμ, with. M N TM N an infinitesimal 0-form Lie algebra valued parameter. The Chern–Simons action, invariant up to boundary terms, is. In (2.8), l is a parameter with units of length (in the natural system of units), necessary in order to take into account the difference between the dimensions of the vielbein eA and of the spin connection ωAB. The parameters k and l are related to the Newton’s constant G (∝ s l2/k) and to the cosmological constant (∝ s/l2) [11]. As we can see we have the presence of the Einstein–Hilbert term and the cosmological constant one, in addition to the first term, which is of the Gauss–Bonnet type
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