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

October 18, 2019

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.

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