Dual Massive Gravity

The linearized massive gravity in three dimensions, over any maximally symmetric background, is known to be presented in a self-dual form as a first order equation which encodes not only the massive Klein-Gordon type field equation but also the supplementary transverse-traceless conditions. We generalize this construction to higher dimensions. The appropriate dual description in d dimensions, additionally to a (non-symmetric) tensor field $h_{\mu\nu}$, involves an extra rank-(d-1) field equivalently represented by the torsion rank-3 tensor. The symmetry condition for $h_{\mu\nu}$ arises on-shell as a consequence of the field equations. The action principle of the dual theory is formulated. The focus has been made on four dimensions. Solving one of the fields in terms of the other and putting back in the action one obtains two other equivalent formulations of the theory in which the action is quadratic in derivatives. In one of these representations the theory is formulated entirely in terms of a rank-2 non-symmetric tensor $h_{\mu\nu}$. This quadratic theory is not identical to the Fierz-Pauli theory and contains the coupling between the symmetric and antisymmetric parts of $h_{\mu\nu}$. Nevertheless, the only singularity in the propagator is the same as in the Fierz-Pauli theory so that only the massive spin-2 particle is propagating. In the other representation, the theory is formulated in terms of the torsion rank-3 tensor only. We analyze the conditions which follow from the field equations and show that they restrict to 5 degrees of freedom thus producing an alternative description to the massive spin-2 particle. A generalization to higher dimensions is suggested.