# Fusion of spin fields in $W_3$ conformal field theories

## TL;DR

This paper identifies a primary field in $W_3$ conformal field theories that generates full $ ext{Z}_3$ sectors through fusion, introducing new fields and providing insights into conformal dimensions relevant for bootstrap bounds.

## Contribution

It introduces a primary field with rational Kac indices in $W_3$ theories that generates complete $ ext{Z}_3$ sectors and identifies associated degenerate fields, extending understanding of fusion processes.

## Key findings

- Identification of a primary field $\sigma$ with rational Kac indices.
- Explicit fusion processes generating $ ext{Z}_3$ sectors.
- Approximate conformal dimension curves for bootstrap bounds.

## 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 $\sigma \times \sigma^*$, 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.