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

A massless graviton in $D$ dimensions has a symmetric traceless transverse polarization tensor. Its state count isThus the metric of eleven-dimensional supergravity has $44$ polarizations, while its three-form has $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$. The total bosonic count is $128$.

For circle reduction to ten dimensions, split $M=(\mu ,10)$. The metric supplies $g_{\mu \nu }$, a Kaluza-Klein vector from $G_{\mu ,10}$, and a scalar from $G_{10,10}$, conventionally the dilaton. The three-form supplies $C_{\mu \nu \rho }=A_{\mu \nu \rho }$ and $B_{\mu \nu }=A_{\mu \nu ,10}$. Their ten-dimensional counts areThis is the bosonic content of type IIA supergravity: $g$, $B_2$ and the dilaton together with $C_1$ and $C_3$. Consequently

For flat seven-torus reduction directly to four dimensions, split $M=(\mu ,i)$, $i=1,\dots ,7$. The metric yields one four-dimensional metric, seven vectors $G_{\mu i}$ and $7\cdot 8/2=28$ scalars $G_{ij}$. Their polarizations are $2+7\cdot 2+28=44$. The three-form givesThe three-form count is $0+7+42+35=84$, as required by toroidal reduction of a p-form gauge field. Applying two-form scalar duality to the seven two-forms, the propagating four-dimensional spectrum isThis is the bosonic content of four-dimensional N=8 supergravity. The remaining four-dimensional three-form has no local polarization and can be omitted from the propagating spectrum, though a flux choice can matter in other backgrounds. The toroidal reduction of eleven-dimensional supergravity here retains all zero modes; fixing moduli, introducing fluxes or using an orbifold projection would change the spectrum and would not be this unprojected counting problem.