Interactions of a single massless tensor field with the mixed symmetry (3,1). No-go results

Non-trivial consistent interactions of a single massless tensor field t_λμν|α with the mixed symmetry corresponding to the two-column Young diagram (3,1), dual to linearized gravity in D = 6 space-time dimensions, are studied in the following cases: self-couplings, cross-interactions with a Pauli-Fierz field and cross-couplings to purely matter theories. Our approach is based on the deformation of the solution to the master equation from the antifield-BRST formalism by means of the local cohomology of the BRST differential. Our main results, obtained under the assumptions that the emerging interactions are local, smooth, Lorentz-covariant, Poincaré-invariant and of maximum derivative order equal to two, can be synthesized into: no consistent self-couplings exist; no cross-interactions with the Pauli-Fierz field can be added; no non-trivial consistent cross-couplings to the matter theories are allowed.