Noncommutative geometry and gravity

Abstract

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered.The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.

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.…”