A gravity theory on noncommutative spaces

Abstract

A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter θ. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra a covariant tensor calculus is constructed and all the concepts like metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a θ-deformed Einstein-Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in θ.

Comments on noncommutative gravity

“…In the case of noncommutative gravity a similar approach has been advanced in [7,8] where local diffeomorphisms are twisted. In the following we review some basic aspects of this approach and study the corrections to the standard gravitational action in the weak field expansion around flat space-time.…”

“…In [7,8] a gravity action is presented and shown to be invariant under deformed diffeomorphisms. In D-dimensions this action reads 6…”

“…Recently, however, a new approach has been proposed in which a deformation of the Einstein-Hilbert action is constructed based on a deformation of the group of diffeomorphisms [7,8] (see [9] for a review). The idea behind it is to replace the diffeomorphism invariance of general relativity by a twisted version of this symmetry [10], i.e.…”

“…One may wonder whether string theory can reproduce the deformed gravity action proposed in [7,8] in some limit. It is known that the strict Seiberg-Witten [12] limit results in a complete decoupling of gravity.…”

“…After that, in Section 3, we will summarize the relevant aspects of the construction of the noncommutative gravity action presented in [7,8]. We move on in Section 4 to study the gravity action induced on the brane in the presence of a B-field to the next-to-leading order in the Seiberg-Witten limit.…”