Abstract:The cross-couplings among several massless spin-two fields (described in the free limit by a sum of Pauli-Fierz actions) in the presence of a Dirac field are investigated in the framework of the deformation theory based on local BRST cohomology. Under the hypotheses of locality, smoothness of the interactions in the coupling constant, Poincaré invariance, (background) Lorentz invariance and the preservation of the number of derivatives on each field, we prove that there are no consistent cross-interactions among different gravitons in the presence of a Dirac field. The basic features of the couplings between a single Pauli-Fierz field and a Dirac field are also emphasized.

The cross-couplings among several Weyl gravitons (described in the free limit by a sum of linearized Weyl actions) in the presence of a scalar field are studied with the help of the deformation theory based on local BRST cohomology. Under the hypotheses of locality, smoothness of the interactions in the coupling constant, Poincaré invariance, (background) Lorentz invariance, and the preservation of the number of derivatives on each field, together with the supplementary assumption that the internal metric defined by the sum of Weyl Lagrangians is positively defined, we prove that there are no consistent cross-interactions among different Weyl gravitons in the presence of a scalar field. The couplings of a single Weyl graviton to a scalar field are also discussed.

The proof of the fact that there are no nontrivial, consistent cross-couplings that can be added between the Weyl graviton and the massless Rarita-Schwinger field is accomplished by means of a cohomological approach, based on the deformation of the solution to the master equation from the antifield-Becchi-Rouet-StoraTyutin (BRST) formalism. The procedure developed here relies on the assumptions of locality, smoothness, (background) Lorentz invariance, Poincaré invariance, and preservation of the number of derivatives with respect to each field (the last hypothesis was made only in antighost number zero).

The proof of the fact that there are no nontrivial, consistent cross‐couplings that can be added between the Weyl graviton and the massless Rarita‐Schwinger field is accomplished by means of a cohomological approach, based on the deformation of the solution to the master equation from the antifield‐Becchi‐Rouet‐Stora‐Tyutin (BRST) formalism. The procedure developed here relies on the assumptions of locality, smoothness, (background) Lorentz invariance, Poincaré invariance, and preservation of the number of derivatives with respect to each field (the last hypothesis was made only in antighost number zero).