= Toroidal reduction of a p-form gauge field
{title2=$A_p\mapsto\binom{D-4}{p-r}\text{ four-dimensional }r\text{-forms}$}
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 $\binom{D-4}{p-r}$ copies and $\binom2r$ local polarizations per copy. Comparing coefficients in $(1+t)^{D-4}(1+t)^2$ gives $\sum_r\binom{D-4}{p-r}\binom2r=\binom{D-2}{p}$, proving polarization matching. Four-dimensional three- and four-form potentials contribute no local propagating states.
// Target: supersymmetry.bigb
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