Classification of Shift-Symmetric No-Scale Supergravities

Models of 4D $\mathcal{N}=1$ supergravity coupled to chiral multiplets with vanishing or positive scalar potential have been denoted as no-scale. Of particular interest in the context of string theory are models which additionally possess a shift-symmetry. In this case there exists a dual description of chiral models in terms of real linear multiplets. We classify all ungauged shift-symmetric no-scale supergravities in both formulations and verify that they match upon dualization. Additionally, we comment on the realizations within effective supergravities descending from string compactifications.