Massive two-form field (${\partial }^m{F}_{mnp}={\mu }^2{A}_{np}$)

An antisymmetric tensor $A_{mn}$ with $F=dA$ obeys ${\partial }^m{F}_{mnp}={\mu }^2{A}_{np}$ for $\mu \ne 0$. Taking a divergence yields ${\partial }^m{A}_{mn}=0$, and the remaining equation is the Klein-Gordon equation for each component. The physical particle polarizations form the exterior square of the massive little group vector representation, of dimension $(D-1)(D-2)/2$. Unlike a two-form gauge field, it has physical longitudinal polarizations.

If ${\partial }_{-}$ is invertible, the divergence constraint of a massive two-form field makes $A_{-i}$ and $A_{ij}$ independent. It gives $A_{+-}=-{\partial }_{-}^{-1}{\partial }_i{A}_{-i}$ and $A_{+i}={\partial }_{-}^{-1}(-{\partial }_{+}{A}_{-i}+{\partial }_j{A}_{ji})$. All independent components obey the Klein-Gordon equation. Their count is $(D-2)+\left(\genfrac{}{}{0ex}{}{D-2}{2}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{2}\right)$.