$N=2$ and $N=4$ SUSY Yang-Mills action on $M^4\times (Z_2\oplus Z_2)$ non-commutative geometry

TL;DR

This paper reformulates $N=2$ and $N=4$ SUSY Yang-Mills actions within non-commutative geometry on a specific product space, interpreting scalars as gauge fields along discrete directions.

Contribution

It presents a novel reformulation of supersymmetric Yang-Mills actions using non-commutative geometry on a product of four-dimensional space and a discrete space.

Findings

Simplifies the representation of scalar fields as gauge fields.

Provides a geometric interpretation of supersymmetric actions.

Shows the reformulation is straightforward within this geometric framework.

Abstract

We show that the $N=2$ and $N=4$ SUSY Yang-Mills action can be reformulated in the sense of non-commutative geometry on $M^4\times (Z_2\oplus Z_2)$ in a rather simple way. In this way the scalars or pseudoscalars are viewed as gauge fields along two directions in the space of one-forms on $Z_2\oplus Z_2$.