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.
Categories: math-ph hep-th math.MP