The $\mathbb{Z}_2$-orbifold of the $\mathcal{W}_3$-algebra

The Zamolodchikov $\mathcal{W}_3$-algebra $\mathcal{W}^c_3$ with central charge $c$ has full automorphism group $\mathbb{Z}_2$. It was conjectured in the physics literature over 20 years ago that the orbifold $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$ is of type $\mathcal{W}(2,6,8,10,12)$ for generic values of $c$. We prove this conjecture for all $c \neq \frac{559 \pm 7 \sqrt{76657}}{95}$, and we show that for these two values, the orbifold is of type $\mathcal{W}(2,6,8,10,12,14)$. This paper is part of a larger program of studying orbifolds and cosets of vertex algebras that depend continuously on a parameter. Minimal strong generating sets for orbifolds and cosets are often easy to find for generic values of the parameter, but determining which values are generic is a difficult problem. In the example of $(\mathcal{W}^c_3)^{\mathbb{Z}_2}$, we solve this problem using tools from algebraic geometry.