Toroidal reduction of a p-form gauge field ($A_p\mapsto \left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)\text{ four-dimensional }r\text{-forms}$)

The zero-mode dimensional reduction on a flat $(D-4)$-torus splits a massless p-form gauge field by the number $r$ of spacetime indices. Each sector has $\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)$ copies and $\left(\genfrac{}{}{0ex}{}{2}{r}\right)$ local polarizations per copy. Comparing coefficients in $(1+t{)}^{D-4}(1+t{)}^2$ gives ${\sum }_r\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)\left(\genfrac{}{}{0ex}{}{2}{r}\right)=\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$, proving polarization matching. Four-dimensional three- and four-form potentials contribute no local propagating states.

// Target: supersymmetry.bigb