Fusion of spin fields in $W_3$ conformal field theories

Abstract

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 $\sigma \times {\sigma }^{\ast }$, 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 $\epsilon$. In analogy with the Virasoro case, the exact curves for conformal dimensions $(h_{\sigma },h_{\psi })$ and $(h_{\sigma },h_{\epsilon })$ are expected to give close estimates for the unitarity bounds in the conformal bootstrap analysis.

18 pages, 6 figures