# Worldline CPT and massless supermultiplets

Article

The action for a massless particle in 4D Minkowski space–time has a worldline-time reversing symmetry corresponding to CPT invariance of the quantum theory. The analogous symmetry of the $\mathscr{N}$-extended superparticle is shown to be anomalous when $\mathscr{N}$ is odd; in the supertwistor formalism this is because a CPT-violating worldline-Chern–Simons term is needed to preserve the chiral $\textbf{U(1)}$ gauge invariance. This accords with the fact that no massless $\mathscr{N}$=1 super-Poincaré irrep is CPT-self-conjugate. There is a CPT self-conjugate supermultiplet when $\mathscr{N}$ is even, but it has $\textbf{2}$$^{\mathscr{N}+1}$ states when $\frac{1}{2}$$\mathscr{N}$ is odd (e.g. the $\mathscr{N}$=2 hypermultiplet) in contrast to just $\textbf{2}$$^{\mathscr{N}}$ when $\frac{1}{2}$$\mathscr{N}$ is even (e.g. the $\mathscr{N}$=4 Maxwell supermultiplet). This is shown to follow from a Kramers degeneracy of the superparticle state space when $\frac{1}{2}$$\mathscr{N}$ is odd.