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Orbifolds of Gaiotto-Rap\v{c}\'ak $Y$-algebras. (arXiv:2208.10037v2 [math.QA] UPDATED)

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The universal two-parameter ${\mathcal W}_{\infty}$-algebra is a classifying
object for vertex algebras of type ${\mathcal W}(2,3,\dots, N)$ for some $N$.
Gaiotto and Rap\v{c}\'ak recently introduced a large family of such vertex
algebras called $Y$-algebras, which includes many known examples such as the
principal ${\mathcal W}$-algebras of type $A$. These algebras admit an action
of $\mathbb{Z}_2$, and in this paper we study the structure of their orbifolds.
Aside from the extremal cases of either the Virasoro algebra or the ${\mathcal
W}_3$-algebra, we show that these orbifolds are generated by a single field in
conformal weight $4$, and we give strong finite generating sets.

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