Supersymmetry and Noncommutative Geometry

TL;DR

This paper extends noncommutative geometry's spectral triple framework to include supersymmetry, demonstrating that the resulting 2-forms naturally contain supersymmetric vector multiplet components, enabling a geometric formulation of supersymmetric Yang-Mills theory.

Contribution

It introduces a supersymmetric extension of spectral triples in noncommutative geometry, linking algebraic structures to supersymmetric field components.

Findings

2-forms contain vector multiplet components of supersymmetry

Constructs a supersymmetric Yang-Mills action within noncommutative geometry

Shows compatibility of supersymmetry with spectral triple framework

Abstract

The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra $\cA$ contains both bosonic and fermionic degrees of freedom. The operator $\cD$ of the spectral triple under consideration is the square root of the Dirac operator und thus the forms of the generalized differential algebra constructed out of the spectral triple are in a representation of the Lorentz group with integer spin if the form degree is even and they are in a representation with half-integer spin if the form degree is odd. However, we find that the 2-forms, obtained by squaring the connection, contains exactly the components of the vector multiplet representation of the supersymmetry algebra. This allows to construct an action for supersymmetric Yang-Mills theory in the framework of noncommutative geometry.