Cross-couplings between a Weyl graviton and a massless conformal gravitino: a local BRST cohomological analysis

The problem of constructing consistent cross-couplings between a Weyl graviton and thea massless conformal gravitino is investigated. The starting points are the free theory obtained as the direct sum of the linearized Weyl action and the free action describing the QQQ-gravitino. The analysis is performed within the antifield-BRST deformation framework, where consistent interactions are characterized as deformations of the solution to the classical master equation. The investigation is further supplemented by specific methods of local BRST cohomology. Under a set of strong assumptions, namely analyticity in the deformation parameter, spacetime locality, preservation of the original Poincaré symmetry, and the requirement that the interaction vertices contain at most four spacetime derivatives, all possible consistent deformations are systematically examined. The local BRST cohomological analysis shows that any candidate cross-interaction between the Weyl graviton and the conformal gravitino is either BRST-trivial or obstructed at higher orders of the deformation procedure. It follows that no nontrivial consistent cross-couplings can be introduced between the fields under consideration in four spacetime dimensions within the class of local deformations satisfying the above hypotheses. The obtained result thus provides a no-go theorem for local interactions between a linearized Weyl graviton and a massless conformal gravitino under the stated assumptions.