Topological five-dimensional Chern--Simons gravity theory in the canonical covariant formalism
The topological five-dimensional Chern-Simons gravity was not previously studied and analyzed in the framework of the group manifold approach for gravity (or supergravity). This powerful method allows one to formulate gauge gravity (or supergravity) theories in any dimension by using only geometrical arguments. The efficiency of the method is made evident when the theory is formulated in more than four dimensions. In this paper, it is shown how the extended canonical covariant formalism (CCF) can be used to describe the dynamics of the topological five-dimensional Chern-Simons gravity. This nonlinear model of gravity contains a Gauss-Bonnet term quadratic in curvature, the usual Einstein term, and a cosmological constant term. From the geometrical Lagrangian density which verifies all the prescriptions of the group manifold approach and by using exterior algebra, the first-order CCF is constructed. This first-order formalism covariant in all its steps allows one to find the equations of motion and the constraints in a very simple way. By means of the toroidal-dimensional reduction process, an U(1) gauge model is obtained, in which the different effective interactions can be analyzed. So, by starting from the five-dimensional Chern-Simons gravity theory which naturally contains a Gauss-Bonnet term, and by using purely geometrical arguments, an interacting model with nonminimal coupling to gravity is constructed. Moreover, the nonlinear interaction modifies the Einstein-Maxwell-dilation theory. This model is more general than the Kaluza-Klein model in which the Gauss-Bonnet term is introduced by hand. On the other hand, the CCF is useful for classical formalisms, but at the quantum level the canonical vierbein formalism (CVF) must be used. In the CVF the Hamiltonian is the true generator of time evolution, therefore the relationship between the CCF and the CVF is also analyzed. By omitting explicit calculation, the true Hamiltonian as generator of time evolutions is given in terms of the first-class constraints which closes the constraints algebra. This is found in complete analogy to what happens in the simple gravity theory.
first-order canonical covariant formalism
five-form Lagrangian density toroidal-dimensional reduction process
scientific article; zbMATH DE number 2036966