Noncommutative Superspace, Supermatrix and Lowest Landau Level

TL;DR

This paper constructs flat noncommutative superspaces using graded Lie algebras, ensuring algebraic consistency and relating these spaces to lowest Landau level systems, advancing the understanding of supersymmetric noncommutative geometry.

Contribution

It introduces a method to derive flat noncommutative superspaces from curved manifolds using graded Lie algebras, with automatic satisfaction of algebraic identities and connections to Landau level physics.

Findings

Constructed noncommutative superspaces for d=2 and d=4.

Ensured Jacobi identities and star product associativity.

Linked noncommutative superspaces to lowest Landau level systems.

Abstract

By using graded (super) Lie algebras, we can construct noncommutative superspace on curved homogeneous manifolds. In this paper, we take a flat limit to obtain flat noncommutative superspace. We particularly consider $d=2$ and $d=4$ superspaces based on the graded Lie algebras $osp(1|2)$, $su(2|1)$ and $psu(2|2)$. Jacobi identities of supersymmetry algebras and associativities of star products are automatically satisfied. Covariant derivatives which commute with supersymmetry generators are obtained and chiral constraints can be imposed. We also discuss that these noncommutative superspaces can be understood as constrained systems analogous to the lowest Landau level system.