Noncommutative Supersymmetry in Two Dimensions

TL;DR

This paper generalizes the ${ m N}=1$ super Euclidean algebra to a noncommutative two-dimensional space, revealing new algebraic structures and representations arising from noncommutative geometry and supersymmetry.

Contribution

It introduces a novel noncommutative superalgebra in two dimensions and constructs its representations using superfields in a noncommutative superspace.

Findings

The superalgebra includes non-vanishing (anti)commutators involving new generators.

A consistent closed algebra with extended generators is established.

Fundamental and adjoint representations of the algebra are derived.

Abstract

Based on an argument for the noncommutativity of momenta in noncommutative directions, we arrive at a generalization of the ${\cal N}=1$ super $E^2$ algebra associated to the deformation of translations in a noncommutative Euclidean plane. The algebra is obtained using appropriate representaions of its generators on the space of superfields in a $D=2, {\cal N}=1$ ``noncommutative superspace.'' We find that the (anti)commutators between several (super)translation generators are no longer vanishing, but involve a new set of generators which together with the (super)translation and rotation generators form a consistent closed algebra. We then analyze the spectrum of this algebra in order to obtain its fundamental and adjoint representations.