Towards a formulation of $f(R)$ supergravity

The supersymmetric completion of $R+R^2$ gravity is known to be equivalent to standard supergravity coupled to two chiral supermultiples with a K\"ahler potential of the no-scale type. The generalization of this programme to $f(R)$ gravity can be carried out in an analogous framework in terms of four chiral multiplets with appropriately chosen superpotential and K\"ahler potential. Although the construction strategy has been known for sometime, we carry out this programme further by setting these theories in a closed form. The general framework considered can accommodate supergravity actions local in the scalar curvature, dual to ordinary $N = 1$ supergravities. Although these are in general plagued by ghosts, the framework studied in this work offers a possibility that these can decouple as can be seen in specific examples.