Dual description of supergravity MacDowell-Mansouri theory

In the context of field theory two elements seem to be necessary to search for strong-weak coupling duality: first, a gauge theory formulation and, second, supersymmetry. For gravitation these two elements are present in MacDowell-Mansouri supergravity. The search for an “effective duality” in this theory presents technical and conceptual problems that we discuss. Nevertheless, by means of a field-theoretical approach, which in the Abelian case coincides with S duality, we exhibit a dual theory, with inverted couplings. This results in a supersymmetric nonlinear $\sigma$ model of the Freedman-Townsend type.