Non-trivial, consistent interactions of a free, massless tensor field t µν|αβ with the mixed symmetry of the Riemann tensor are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings with purely matter theories. The main results, obtained from BRST cohomological techniques under the assumptions on smoothness, locality, Lorentz covariance and Poincaré invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, can be synthesized into: no consistent self-couplings exist, but a cosmologicallike term; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings with the matter theories such that the matter fields gain gauge transformations are allowed.

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Consistent Hamiltonian interactions that can be added to an abelian free BF-type class of theories in any n ≥ 4 spacetime dimensions are constructed in the framework of the Hamiltonian BRST deformation based on cohomological techniques. The resulting model is an interacting field theory in higher dimensions with an open algebra of on-shell reducible first-class constraints. We argue that the Hamiltonian couplings are related to a natural structure of Poisson manifold on the target space.

The basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail. It is shown that any non-trivial co-cycle from the local BRST cohomology group can be taken to stop at antighost number three, its last component belonging to the cohomology of the exterior longitudinal derivative and containing non-trivial elements from the (invariant) characteristic cohomology.

The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL (D, R), corresponding to a rectangular, two-column Young diagram with k > 2 rows are studied in detail. In particular, it is shown that any nontrivial co-cycle from the local BRST cohomology group H (s|d) can be taken to stop either at antighost number (k + 1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H (γ) and containing non-trivial elements from the (invariant) characteristic cohomology H inv (δ|d).

Under the hypotheses of smoothness in the coupling constant, locality, Lorentz covariance, and Poincaré invariance of the deformations, combined with the preservation of the number of derivatives on each field, the consistent interactions between a single free massless tensor gauge field with the mixed symmetry of a two-column Young diagram of the type (3,1) and one Abelian vector field have been investigated. The computations are done with the help of the deformation theory based on a cohomological approach, in the context of the antifield-BRST formalism. The main result is that there exist nontrivial cross-couplings between these types of fields in five spatiotemporal dimensions, which break the PT invariance and allow for the deformation of the gauge transformations of the vector field, but not of the gauge algebra.

The local and manifestly covariant Lagrangian interactions in four spacetime dimensions that can be added to a "free" model that describes a generic matter theory and an abelian BF theory are constructed by means of deforming the solution to the master equation on behalf of specific cohomological techniques.

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