Spin-1 duality in $D$ dimensions

It is known that the Maxwell theory in $D$ dimensions can be written in a first-order form (in derivatives) by introducing a totally antisymmetric field which leads to a ($D-3$)-form dual theory. Remarkably, one can replace the antisymmetric field by a symmetric rank-two tensor ($W_{\mu \nu }=W_{\nu \mu }$). Such master action establishes the duality between the Maxwell theory and a fourth-order higher-rank model in a $D$-dimensional flat space-time. A naive generalization to the curved space shows a connection between the recently found $D=4$ critical gravity and the Maxwell-theory plus a coupling term to the Ricci tensor ($R_{\mu \nu }{A}^{\mu }{A}^{\nu }$). The mass of the spin-1 particle which appears in the $D=4$ critical gravity linearized around anti-de Sitter space is the same one obtained from the Ricci coupling term. We also work out, in flat space-time, the explicitly massive case (Maxwell-Proca) which is dual to a second-order theory for $W_{\mu \nu }$.