Four-dimensional gravity on a covariant noncommutative space

Abstract

We formulate a model of noncommutative four-dimensional gravity on a covariant fuzzy space based on SO(1,4), that is the fuzzy version of the dS 4. The latter requires the employment of a wider symmetry group, the SO(1,5), for reasons of covariance. Addressing along the lines of formulating four-dimensional gravity as a gauge theory of the Poincaré group, spontaneously broken to the Lorentz, we attempt to construct a four-dimensional gravitational model on the fuzzy de Sitter spacetime. In turn, first we consider the SO(1,4) subgroup of the SO(1,5) algebra, in which we were led to, as we want to gauge the isometry part of the full symmetry. Then, the construction of a gauge theory on such a noncommutative space directs us to use an extension of the gauge group, the SO(1,5)×U(1), and fix its representation. Moreover, a 2-form dynamic gauge field is included in the theory for reasons of covariance of the transformation of the field strength tensor. Finally, the gauge theory is considered to be spontaneously broken to the Lorentz group with an extension of a U(1), i.e. SO(1,3)×U(1). The latter defines the four-dimensional noncommutative gravity action which can lead to equations of motion, whereas the breaking induces the imposition of constraints that will lead to expressions relating the gauge fields. It should be noted that we use the Euclidean signature for the formulation of the above programme.

Gauge theory of gravity on a four-dimensional covariant noncommutative space

“…Targeting minimal extension of the symmetry, we are led to adopt the SO(1, 5) group. Therefore, a fuzzy version of dS 4 space, respecting covariance, is obtained after the symmetry is enlarged to the SO(1, 5) [45]. To facilitate the whole construction and the calculations we adopt the Euclidean signature, which means that instead of the SO(1, 5), the symmetry group we consider is that of SO( 6).…”

“…However, simply by counting the degrees of freedom, adoption of the above breaking would lead to an overconstrained theory. For this reason, we perform the symmetry breaking imposing less straightforward constraints [45]. In turn, the first condition is the torsionless condition: which is also encountered in the cases of the Einstein and conformal gravity theories when described as gauge theories.…”

“…However, the existence of "covariant noncommutative spaces" [36,37] which preserve the Lorentz invariance, allows the construction of field theories from noncommutative deformations. [38,39,40,41,42,43,44,45,46,47] (see also [48,49,50,51,52]).…”

“…The purpose of this article is to present and highlight the features of a four-dimensional noncommutative gravity model, constructed recently [45] as a gauge theory on a fuzzy space, the dS 4 . Drawing motivation from the work of Heckman-Verlinde [37], who were based on Yang's early publication [36], we have constructed a covariant fuzzy dS space with the property of preserving Lorentz invariance.…”

“…A Yang-Mills action of a theory on the fuzzy dS 4 space is gauge invariant. For details see Appendix A of the first paper of[45].…”