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.