Comments of $\mathbb{Z}_2^2$-supersymmetry in superfield formalism

TL;DR

This paper explores the superfield formulation of minimal $bZ_2^2$-supersymmetry, demonstrating how integrability in superspace ensures action invariance and classifying superfields into irreducible representations.

Contribution

It introduces a superfield formalism for $bZ_2^2$-supersymmetry, analyzing representation theory and identifying conditions for invariant actions.

Findings

Integrability guarantees invariance of the action.

Superfields with different $bZ_2^2$-degrees form irreducible representations.

Not all superfields lead to integrable, invariant Lagrangians.

Abstract

We investigate superfield formulation of the minimal $\mathbb{Z}_2^2$-supersymmetry. It is shown that the integrability on $\mathbb{Z}_2^2$-superspace guarantees the invariance of action. Then we present two superfields which carry distinct irreducible representations of the $\mathbb{Z}_2^2$-supersymmetry algebra. One of them gives integrable Lagrangian and the other does not. We also show that integrable superfields with different $\mathbb{Z}_2^2$-degree also carry irreducible representations and they give invariant actions. To perform this analysis, the representation theory of the minimal $\mathbb{Z}_2^2$-supersymmetry algebra is studied in some detail.