Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 53 / 3 / b / Solution

For a null momentum, use light cone gauge to remove potential components along one null direction; the free field equations remove the remaining longitudinal components. The physical polarizations of a massless p-form gauge field are therefore antisymmetric tensors on the $D-2$ transverse directions, transforming under the rotational little group $SO(D-2)$. Choosing $p$ distinct transverse indices givesThis already takes account of the reducibility of the higher-form gauge parameter; naïvely subtracting just one gauge-parameter component count would not give the right answer. No self-duality constraint is being imposed.

For dimensional reduction to four dimensions, retain the zero modes on a flat $(D-4)$-torus, without fluxes or projections. Write $d=D-4$ and split tensor indices into spacetime indices $\mu$ and internal indices $i$. A component with $r$ spacetime indices is a four-dimensional $r$-form, with $p-r$ internal indices and hence multiplicity $\left(\genfrac{}{}{0ex}{}{d}{p-r}\right)$. The complete field decomposition isUse the convention that a binomial coefficient vanishes when its lower entry lies outside its range. In four dimensions the scalar, vector and two-form have respectively $1,2,1$ local polarizations, while the three- and four-form potentials have none. Hence toroidal reduction of a p-form gauge field givesThe middle identity follows by comparing the coefficient of $t^p$ in $(1+t{)}^2(1+t{)}^d=(1+t{)}^{d+2}$. ThereforeA four-dimensional two-form may instead be counted as one scalar by two-form scalar duality, with the same result. Higher-form zero modes can describe nondynamical flux data, which are distinct from these propagating degrees of freedom. Nonzero Kaluza-Klein modes form massive lower-dimensional multiplets with longitudinal components supplied by the associated gauge fields; the displayed matching concerns the massless zero-mode sector.