By writing the theory of a partially massless spin-s field in a first order formalism, and defining the associated generalised curvature, one can construct dualities which are analogous to the electric-magnetic duality in massless spin-1 D=4. A symmetric s-index gauge field is dual to another partially massless s-index gauge field, for all spins s and depths t.

On curved backgrounds, the underlying (A)dS group has spin-s representations for which gauge invariances remove 2t+1 of the 2s+1 helicity components. These are called partially massless fields of depth t. For massless spin-1 fields, we have a gauge invariant Maxwell field strength and an electromagnetic duality. For massless spin-2, we analogously have the Riemann curvature tensor and dual (Weyl tensor). Here, the authors have generalised to partially massless fields of arbitrary spin and depth, defining generalised curvatures and dualities.

Previously, 1306.1835 has proposed higher-derivative matter fields, and can be written via non-dynamical auxiliary fields. In such a theory, there are two free parameters. Demanding stability of a Minkowski vacuum, as well as the absence of tachyons, places bounds on these parameters.

One can interpret such non-conventional higher-derivative matter fields as a modification to gravity. These considerations restrict this possible modification to GR. The two bounds derived here are necessary, but by no mean sufficient - the parameter space of these higher-derivative theories is potentially very restricted.