Fusion of spin fields in $W_3$ conformal field theories

In generic conformal field theories with $W_3$ symmetry, we identify a primary field $sigma$ with rational Kac indices, which produces the full $mathbb{Z}_3$ charged and neutral sectors by the fusion processes $sigma times sigma$ and $sigma times sigma^*$, respectively. In this sense, this field generalises the $mathbb{Z}_3$ fundamental spin field of the three-state Potts model. Among the degenerate fields produced by these fusions, we single out a `parafermion field $psi$ and an `energy field $varepsilon$. In analogy with the Virasoro case, the exact curves for conformal dimensions $(h_sigma,h_psi)$ and $(h_sigma,h_varepsilon)$ are expected to give close estimates for the unitarity bounds in the conformal bootstrap analysis.