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.

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 the only consistent interactions in D = 11 among massless gravitini, a graviton, and a 3-form are described by N = 1, D = 11 SUGRA.

Under the general hypotheses of locality, smoothness of interactions in the coupling constant, Poincaré invariance, Lorentz covariance, and preservation of the number of derivatives on each field, we investigate the cross-couplings of one or several spin-two fields to a massless p-form. Two complementary cases arise. The first case is related to the standard interactions from General Relativity, but the second case describes a new, special type of couplings in D = p + 2 spacetime dimensions, which break the PT-invariance. Nevertheless, no consistent, indirect cross-interactions among different gravitons with a positively defined metric in internal space can be constructed.