Supersymmetry and noncommutative geometry Part I: Supersymmetric almost-commutative geometries

TL;DR

This paper investigates the potential for supersymmetry within noncommutative geometry frameworks, focusing on almost-commutative geometries on flat 4D backgrounds, and identifies the limited conditions under which supersymmetric actions can arise.

Contribution

It provides a systematic analysis of supersymmetry possibilities in almost-commutative geometries, introducing a method to identify building blocks for supersymmetric theories.

Findings

Examples satisfying supersymmetry conditions are rare.

A new approach to identify supersymmetric building blocks is proposed.

The analysis is limited to flat 4-dimensional backgrounds.

Abstract

Noncommutative geometry has seen remarkable applications for high energy physics, viz. the geometrical interpretation of the Standard Model. The question whether it also allows for supersymmetric theories has so far not been answered in a conclusive way. In this first of three papers we do a systematic analysis of the possibilities for almost-commutative geometries on a 4-dimensional, flat background to exhibit not only a particle content that is eligible for supersymmetry but also have a supersymmetric action. We come up with an approach in which we identify the basic 'building blocks' of potentially supersymmetric theories and the demands for their action to be supersymmetric. Examples that satisfy these demands turn out to be sparse.