# Four-dimensional Gravity on a Covariant Noncommutative Space

## TL;DR

This paper develops a covariant noncommutative four-dimensional gravity model on a fuzzy de Sitter space using extended gauge groups and spontaneous symmetry breaking, advancing noncommutative geometry in gravitational theories.

## Contribution

It introduces a novel noncommutative gravity model based on SO(1,4) and SO(1,5) gauge groups with a U(1) extension, incorporating a 2-form gauge field for covariance.

## Key findings

- Constructed a gauge theory on fuzzy de Sitter space with extended symmetry groups.
- Derived equations of motion and constraints from symmetry breaking.
- Proposed a noncommutative gravity action compatible with covariance.

## 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 $\text{dS}_4$. 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 Poincar\'e 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)$\times$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)$\times$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.