We explore the possibility of parity-violating, nonminimally coupled two-form field theories that retain the same dynamical degrees of freedom as a massive two-form and thus are ghost-free. Starting from the most general kinetic terms and dimension-four couplings between the two-form field and the curvature tensors, we find a two-parameter family of such theories. However, we also find that parity-violating terms involving the dual two-form field can be absorbed into a field redefinition, leaving a theory with essentially the same structure as that obtained by Heisenberg and Trenkler (i.e., the parity-preserving coupling to the double dual Riemann tensor). After the field redefinition, the only place where parity-violating terms can appear is in the potential. We then consider a homogeneous and isotropic cosmological model with an isotropic configuration of a triplet of two-form fields, and study tensor perturbations in this setup. There are three types of tensor perturbations, two of which are dynamical. We show that chiral gravitational waves can be generated in the presence of parity-violating terms in the potential.

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• Alternative gravity theories