Non-linearly ghost-free higher curvature gravity

We find unitary and local theories of higher curvature gravity in the vielbein formalism, known as the Poincar\'{e} gauge theory by utilizing the equivalence to the ghost-free massive bigravity. We especially focus on three and four dimensions but extensions into a higher dimensional spacetime are straightforward. In three dimensions, a 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 non-linear orders. In a special limit, new massive gravity is recovered. When the model is applied to the AdS/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 ghost at non-linear orders requires an infinite number of higher curvature terms, and these terms can be given by a schematic form $R(1+R/\alpha m^2)^{-1}R$ where $m$ is the mass of the massive spin-2 mode originating from the higher curvature terms and $\alpha$ 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 the massive spin-2 mode.