On higher derivatives in 3D gravity and higher-spin gauge theories

The general second-order massive field equations for arbitrary positive integer spin in three spacetime dimensions, and their ‘‘self-dual” limit to first-order equations, are shown to be equivalent to gauge-invariant higher-derivative field equations. We recover most known equivalences for spins 1 and 2, and find some new ones. In particular, we find a non-unitary massive 3D gravity theory with a 5th order term obtained by contraction of the Ricci and Cotton tensors; this term is part of an N = 2 super-invariant that includes the ‘‘extended Chern-Simons” term of 3D electrodynamics. We also find a new unitary 6th order gauge theory for ‘‘self-dual” spin 3.