Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 53 / 4 / d / Solution

In extended supersymmetry there are $\mathcal{N}$ Weyl-spinor charges $Q_{\alpha }^A$, $A=1,\dots ,\mathcal{N}$. A positive Hermitian mixed bracket can be diagonalized and normalized in this internal index. The ordinary four-dimensional particle algebra isAll supercharges commute with translations. The new central charges in supersymmetry are Lorentz scalars commuting with momentum, Lorentz generators and supercharges. Their antisymmetry follows because the full anticommutator is symmetric under interchange of the two paired spinor/internal labels, while ${ϵ}_{\alpha \beta }$ is antisymmetric. They can be nonzero only for $\mathcal{N}>1$. Internal R-symmetry automorphisms rotate the supercharge index; the algebra without specified central charges admits unitary rotations, and a chosen central-charge matrix restricts the subgroup preserving it. Central charges allow BPS states and shortened massive representations, with the numerical BPS normalization depending on whether a factor two is included in the definition of $Z$.

For a massless unitary representation at the momentum used above, every $Q_1^A$ has zero anticommutator with its adjoint and hence vanishes. Its mixed same-chirality bracket with $Q_2^B$ then forces $Z^{AB}=0$ on that representation. The surviving $\mathcal{N}$ charges give $\mathcal{N}$ independent fermionic creation operators, each lowering helicity by one half. Starting from helicity $\lambda$, their antisymmetric products giveThus the helicity spectrum of a massless supermultiplet occupies an interval of width $\mathcal{N}/2$ and has $2^{\mathcal{N}}$ states before any separate CPT completion.

The assumption excluding helicity above two also excludes helicity below minus two in a CPT-complete physical spectrum. Therefore both ends of the interval must obeyand the spin bound for massless supermultiplets givesThis bound is attained: for $\mathcal{N}=8$ and highest helicity two, the helicities are $2,3/2,1,1/2,0,-1/2,-1,-3/2,-2$ with multiplicities $1,8,28,56,70,56,28,8,1$. They form the CPT-self-conjugate multiplet of four-dimensional N=8 supergravity, with $128$ bosonic and $128$ fermionic states. Its bosonic graviton, $28$ vectors and $70$ scalars agree with the toroidal reduction counted above. Here $\mathcal{N}=8$ means eight Weyl-spinor supersymmetries, or $32$ real supercharge components, not eight real components.