Airy ideals, transvections and $\mathcal{W}(\mathfrak{sp}_{2N})$-algebras

In the first part of the paper we propose a different viewpoint on the theory of higher Airy structures (or Airy ideals) which may shed light on its origin. We define Airy ideals in the $\hbar$-adic completion of the Rees Weyl algebra, and show that Airy ideals are defined exactly such that they are always related to the canonical left ideal generated by derivatives by automorphisms of the Rees Weyl algebra of a simple type, which we call transvections. The standard existence and uniqueness result in the theory of Airy structures then follows immediately. In the second part of the paper we construct Airy ideals generated by the non-negative modes of the strong generators of the principal $\mathcal W$-algebra of $\mathfrak{sp}_{2N}$ at level $N-1/2$, following the approach developed in arXiv:1812.08738. This provides an example of an Airy ideal in the Heisenberg algebra that requires realizing the zero modes as derivatives instead of variables, which leads to an interesting interpretation for the resulting partition function.