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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Cross-couplings between a massless spin-two field (described in the free limit by the Pauli-Fierz action) and an Abelian three-form gauge field in D = 11 are investigated in the framework of the deformation theory based on local BRST cohomology. These consistent interactions are obtained on the grounds of smoothness in the coupling constant, locality, Lorentz covariance, Poincaré invariance, and the presence of at most two derivatives in the interacting Lagrangian. Our results confirm the uniqueness of the eleven-dimensional interactions between a graviton and a three-form prescribed by General Relativity.

Under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincaré invariance, Lorentz covariance and the preservation of the number of derivatives on each field in the Lagrangian of the interacting theory (the same number of derivatives like in the free Lagrangian), we prove that in D = 11 there are no cross-interactions between the graviton and the massless gravitino and also no self-interactions in the Rarita-Schwinger sector. A comparison with the case D = 4 is briefly discussed.

The interactions that can be introduced between a massless Rarita-Schwinger field and an Abelian three-form gauge field in eleven spacetime dimensions are analyzed in the context of the deformation of the "free" solution of the master equation combined with local BRST cohomology. Under the hypotheses of smoothness of the interactions in the coupling constant, locality, Poincaré invariance, Lorentz covariance, and the presence of at most two derivatives in the Lagrangian of the interacting theory (the same number of derivatives like in the free Lagrangian), we prove that there are neither cross-couplings nor self-interactions for the gravitino in D = 11. The only possible term that can be added to the deformed solution to the master equation is nothing but a generalized Chern-Simons term for the three-form gauge field, which brings contributions to the deformed Lagrangian, but does not modify the original, Abelian gauge transformations.

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.

All consistent interactions in five spacetime dimensions that can be added to a free BF-type model involving one scalar field, two types of one-forms, two sorts of two-forms, and one threeform are investigated by means of deforming the solution to the master equation with the help of specific cohomological techniques. The couplings are obtained on the grounds of smoothness, locality, (background) Lorentz invariance, Poincaré invariance, and the preservation of the number of derivatives on each field.Since a is a d-closed modulo γ p-form, the equations (148) and (163) yieldWith the help of the property (155), from (164) we arrive toTheorem A.1 then implies thatwith β J an invariant polynomial. Inserting α J of the form (166) in (153), we obtain thatDue to the fact that de J = γê J and γλ 4 I,J = 0, we consequently have thatIn a similar manner, with the help of the relation (276) inserted in (271), we arrive to b I−1 = s ± Now, if we simultaneously perform some trivial redefinitions of a I and of the 'current' b I−1 like a ′ I = a I − s ± J λ 5 I+1and, meanwhile, fixā I andb I−1 from (267) and respectively from (271) as a I = ± J λ 4 I,Jê J , (282) b I−1 = ± J λ 3 I−1,Jê J ,