Generalized AdS-Lorentz deformed supergravity on a manifold with boundary

The purpose of this paper is to explore the supersymmetry invariance of a particular supergravity theory, which we refer to as D=4 generalized AdS-Lorentz deformed supergravity, in the presence of a non-trivial boundary. In particular, we show that the so-called generalized minimal AdS-Lorentz superalgebra can be interpreted as a peculiar torsion deformation of osp(4|1), and we present the construction of a bulk Lagrangian based on the aforementioned generalized AdS-Lorentz superalgebra. In the presence of a non-trivial boundary of space-time, that is when the boundary is not thought of as set at infinity, the fields do not asymptotically vanish, and this has some consequences on the invariances of the theory, in particular on supersymmetry invariance. In this work, we adopt the so-called rheonomic (geometric) approach in superspace and show that a supersymmetric extension of a Gauss-Bonnet like term is required in order to restore the supersymmetry invariance of the theory. The action we end up with can be recast as a MacDowell-Mansouri type action, namely as a sum of quadratic terms in the generalized AdS-Lorentz covariant super field-strengths.