Critical and non-critical Einstein-Weyl supergravity

Abstract

We construct \( \mathcal{N} = 1 \) supersymmetrisations of some recently-proposed theories of critical gravity, conformal gravity, and extensions of critical gravity in four dimensions. The total action consists of the sum of three separately off-shell supersymmetric actions containing Einstein gravity, a cosmological term and the square of the Weyl tensor. For generic choices of the coefficients for these terms, the excitations of the resulting theory around an AdS4 background describe massive spin-2 and massless spin-2 modes coming from the metric; massive spin-1 modes coming from a vector field in the theory; and massless and massive spin-\( \frac{3}{2} \) modes (with two unequal masses)coming from the gravitino. These assemble into a massless and a massive \( \mathcal{N} = 1 \) spin-2 multiplet. In critical supergravity, the coefficients are tuned so that the spin-2 mode in the massive multiplet becomes massless. In the extensions of critical supergravity, the coefficients are chosen so that the massive modes lie in a “window” of lowest energies E 0 such that these ghostlike fields can be truncated by imposing appropriate boundary conditions at infinity, thus leaving just positive-norm massless supergravity modes.