Weak supersymmetry

TL;DR

This paper investigates weak supersymmetric systems in low dimensions, constructing a nontrivial interacting example in one dimension that extends traditional supersymmetry by including extra bosonic generators.

Contribution

It introduces and constructs a nontrivial interacting weak supersymmetric system in one dimension, expanding the understanding of supersymmetry beyond standard frameworks.

Findings

Constructed a nontrivial interacting weak supersymmetric system in 1D.

Connected weak supersymmetry to n-fold supersymmetric and quasi-exactly solvable systems.

Showed that traditional no-go theorems do not apply in 1D for such systems.

Abstract

We explore ``weak'' supersymmetric systems whose algebra involves, besides Poincare generators, extra bosonic generators not commuting with supercharges. This allows one to have inequal number of bosonic and fermionic 1--particle states in the spectrum. Coleman--Mandula and Haag--Lopuszanski--Sohnius theorems forbid the presence of such extra bosonic charges in {\it interacting} theory for $d \geq 3$. However, these theorems do not apply in one or two dimensions. For $d=1$, we construct a nontrivial interacting system characterized by weak supersymmetric algebra. It is related to ``n--fold'' supersymmetric systems and to quasi-exactly solvable systems studied earlier.