I'll describe an old-fashioned way to quantize an energy level of a complexified 1d configuration space. A quantum energy level turns out to be a CP^1 structure on the configuration space, and its perturbations are deformations of that geometric structure. This offers a geometric perspective on one of the algebraic perturbation theory results that underpin the exact WKB method.
Airy structures - a notion introduced by Maxim Kontsevich and Yan Soibelman - are certain systems of linear ODEs admitting a unique solution (the so-called partition function) which is computed by a topological recursion. Airy structure therefore serve as initial data for the recursion. I will review the strategy to obtain Airy structures from the W(gl_r) vertex operator algebra (from joint work with Bouchard, Chidambaram, Creutzig, Noshchenko). This strategy depends on the choice of an element of the Weyl group of gl_r, used for "twisting". For a Coxeter element this is equivalent to Bouchard-Eynard topological recursion on spectral curves with higher order ramification. For other twisting we show with Aghaei, Kramer and Schueler that this defines a topological recursion on singular spectral curves. Schueler has classified which twisting do lead to Airy structures: surprisingly there are some restrictions, posing non-trivial conditions on the type of spectral curves allowed in this definition, and whose interpretation is still mysterious.