Nonlinearly ghost-free higher curvature gravity

Abstract

We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as Poincar\'e gauge theory, by utilizing the equivalence to ghost-free massive bigravity. We especially focus on three and four dimensions, but extensions into a higher-dimensional spacetime are straightforward. In three dimensions, quadratic gravity $\mathcal{L}=R+{T}^{2}+{R}^{2}$, where $R$ is the curvature and $T$ is the torsion with indices omitted, is shown to be equivalent to zwei-dreibein gravity and free from the ghost at fully nonlinear orders. In a special limit, new massive gravity is recovered. When the model is applied to the $\mathrm{AdS}/\mathrm{CFT}$ correspondence, unitarity both in the bulk theory and in the boundary theory implies that the torsion must not vanish. On the other hand, in four dimensions, the absence of a ghost at nonlinear order requires an infinite number of higher curvature terms, and these terms can be given by a schematic form $R(1+R/\ensuremath{\alpha}{m}^{2}{)}^{\ensuremath{-}1}R$, where $m$ is the mass of the massive spin-2 mode originating from the higher curvature terms and $\ensuremath{\alpha}$ is an additional parameter that determines the amplitude of the torsion. We also provide another four-dimensional ghost-free higher curvature theory that contains a massive spin-0 mode as well as a massive spin-2 mode.