Discrete Symmetry, Non-Commutative Geometry and Gravity

TL;DR

This paper explores a geometric framework combining scalar fields, gravity, and non-commutative geometry using differential graded algebras, leading to a model of two coupled universes with extended fields.

Contribution

It introduces a novel formalism using Z_2-graded algebras to incorporate scalar and vector fields into gravity, extending non-commutative geometric approaches.

Findings

Describes a new geometric model with coupled universes.

Extends the connection and vierbeins to include additional fields.

Proposes a different perspective from standard non-commutative geometry in gravity.

Abstract

We describe the geomety of a set of scalar fields coupled to gravity. We consider the formalism of a differential Z_2-graded algebra of $2\times 2$ matrices whose elements are differential forms on space-time. The connection and the vierbeins are extended to incorporate additional scalar and vector fields. The resulting action describes two universes coupled in a non-minimal way to a set of scalar fields. This picture is slightly different from the description of general relativity in the framework of non-commutative geomety.