Massless supermultiplet

A massless supermultiplet is a positive-energy unitary irreducible representation of the Super-Poincaré algebra with null four-momentum. For finite helicity states in four dimensions, half the supercharges act trivially and the others form a fermionic Fock space.

For a four-dimensional $\mathcal{N}=1$ finite-helicity massless supermultiplet, one complex oscillator changes helicity by one-half. A lower-helicity labelling convention assigns $\kappa$ to the pair $(\kappa ,\kappa +1/2)$. A symmetric midpoint convention instead uses $\kappa +1/4$; report the helicity pair when comparing conventions. CPT completion of a supermultiplet pairs $\kappa$ with $-\kappa -1/2$.

For a four-dimensional massless supermultiplet with $\mathcal{N}$ independent supercharges and highest helicity $\lambda$, level $k$ has helicity $\lambda -k/2$ and multiplicity $\left(\genfrac{}{}{0ex}{}{\mathcal{N}}{k}\right)$. The total is $2^{\mathcal{N}}$ states and the helicity range has width $\mathcal{N}/2$, before any necessary CPT completion of a supermultiplet.

In four-dimensional supersymmetry, a finite-helicity massless supermultiplet has helicity interval of width $\mathcal{N}/2$. If every state obeys $|h|\le s$, that interval fits inside one of width $2s$, giving $\mathcal{N}\le 4s$. The bounds $4$ for $s=1$ and $8$ for $s=2$ are attained by the four-dimensional N=4 super Yang-Mills theory vector multiplet and the four-dimensional N=8 supergravity supergravity multiplet. Spin bounds alone do not prove that an arbitrary action is renormalizable.