A note on the ubiquity of the traceless spin connection in dual massive spin-2 theories

There are three families of second-order (in derivatives) models for massive spin-2 particles via a nonsymmetric rank-2 tensor. Each family depends upon an arbitrary real parameter. One of the families includes the usual Fierz-Pauli (FP) model. By starting from a dual formulation of the FP theory in terms of a spin connection it is known that one can derive a fourth-order model for massive 'spin-2' particles in arbitrary dimensions. It is a higher dimensional version of the linearized 3D New Massive Gravity. Here, we show that the same approach applied for the other nonsymmetric rank-2 families leads exactly to the same fourth-order theory. We point out that this is a consequence of the ubiquity of a traceless spin connection in the dual formulation. The Weyl symmetry in the massless limit of one of the rank-2 families is behind the appearance of the traceless spin connection in the dual model. We derive dual maps between correlation functions in the rank-2 (vielbein) and rank-3 (traceless spin connection) dual formulations.