Stability of the 3-form field during inflation

We consider the minimally coupled 3-form field, which has been considered as a candidate for realizing inflation. We have studied the conditions to avoid ghosts and Laplacian instabilities and found that some classes of potentials, e. g., the Mexican-hat, will in general be unstable. We then propose other classes of potentials which are instead free from any instability, drive a long enough slow-roll regime followed by an oscillatory epoch, and as a consequence, can provide successful inflation. Finally, we also provide stable potentials that lead to a small enough propagation speed for the scalar perturbations, making it possible for these models to produce non-Gaussianities.