Gravitational fields on a noncommutative space

TL;DR

This paper extends noncommutative gravity theories to even-dimensional spaces, proposing an action framework and analyzing specific cases like noncommutative ${f CP}^2$ and $S^4$.

Contribution

It introduces a new action formulation for gravitational fields on even-dimensional noncommutative spaces, expanding beyond the three-dimensional case.

Findings

Formulated an action for noncommutative gravity on ${f CP}^2$ and $S^4$

Extended noncommutative gravity from 3D to even dimensions

Provided detailed analysis of the proposed action in specific geometries

Abstract

Noncommutative three-dimensional gravity can be described in terms of a noncommutative Chern-Simons theory. We extend this structure and also propose an action for gravitational fields on an even dimensional noncommutative space. The action is worked out in some detail for fields on a noncommutative ${\bf CP}^2$ and on $S^4$.