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

The Zamolodchikov ${\mathcal{W}}_3$-algebra ${\mathcal{W}}_3^c$ 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 $({\mathcal{W}}_3^c{)}^{{\mathbb{Z}}_2}$ 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 $({\mathcal{W}}_3^c{)}^{{\mathbb{Z}}_2}$, we solve this problem using tools from algebraic geometry.