Noncommutative Geometry and Gravity

TL;DR

This paper develops a covariant differential geometry framework on noncommutative spaces using a twist deformation, aiming to formulate Einstein's gravity equations in a noncommutative setting.

Contribution

It introduces a general twist-based deformation of diffeomorphisms and constructs a covariant, coordinate-independent differential geometry on noncommutative manifolds, including non-anticommutative superspaces.

Findings

Established a star-product-based differential geometry on noncommutative spaces

Demonstrated covariance under deformed diffeomorphisms

Set the stage for formulating Einstein's equations in noncommutative geometry

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.