Abstract
Abstract We study in more detail the cubic constraints for $${\mathcal N}=1$$ N = 1 chiral superfields proposed in the earlier work Eur. Phys. J. C 81, 523 (2021), which describe low-energy goldstino-axion dynamics in global non-linearly realized supersymmetry. We generalize the constraint (i) by gauging the abelian symmetry that shifts the axion, and (ii) to supergravity. In the former case we show that after adding the abelian gauge multiplet, the construction can be formulated as a cubic nilpotent massive vector superfield. In the latter case we find explicit solution to the constraint including contributions from the supergravity multiplet. We also propose microscopic supergravity models (with linear supersymmetry), including a toy model for the string dilaton, which give rise to the cubic constraints at low energies.