Poincar\'e Supersymmetry Representations Over Trace Class Noncommutative Graded Operator Algebras

TL;DR

This paper extends four-dimensional supersymmetry theories to trace class noncommutative operator algebras, enabling their reformulation as supersymmetric matrix models with detailed component and superspace analyses.

Contribution

It introduces a novel framework for representing supersymmetry via trace supercharges in noncommutative graded operator algebras, connecting field theories to matrix models.

Findings

Supersymmetry can be realized through trace supercharges in noncommutative algebras.

The approach applies to Wess-Zumino and supersymmetric Yang-Mills models.

Component and superspace calculations support the theoretical framework.

Abstract

We show that rigid supersymmetry theories in four dimensions can be extended to give supersymmetric trace (or generalized quantum) dynamics theories, in which the supersymmetry algebra is represented by the generalized Poisson bracket of trace supercharges, constructed from fields that form a trace class noncommutative graded operator algebra. In particular, supersymmetry theories can be turned into supersymmetric matrix models this way. We demonstrate our results by detailed component field calculations for the Wess-Zumino and the supersymmetric Yang-Mills models (the latter with axial gauge fixing), and then show that they are also implied by a simple and general superspace argument.