Unitary representations of the $\mathcal{W}_3$-algebra with $c\geq 2$

TL;DR

This paper proves the unitarity of the vacuum representation of the $ ext{W}_3$-algebra for all central charges $c extgreater= 2$, constructing explicit unitary representations and clarifying unitarity conditions for lowest weight modules.

Contribution

It establishes unitarity of the $ ext{W}_3$-algebra's vacuum representation for all $c extgreater= 2$ and explicitly constructs many unitary representations, extending previous results.

Findings

Unitarity holds for all $c extgreater= 2$ in the vacuum representation.

Explicit construction of unitary representations for many positive lowest weight values.

Complete classification of unitarity for irreducible lowest weight representations in $2 extless c extless 98$.

Abstract

We prove unitarity of the vacuum representation of the $\mathcal{W}_3$-algebra for all values of the central charge $c\geq 2$. We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the $\mathcal{W}_3$-algebra in the $2\leq c\leq 98$ region.