Higher Airy structures and topological recursion for singular spectral curves

TL;DR

This paper classifies quantum Airy structures linked to $W(rak{gl}_r)$-algebras, extending topological recursion to singular spectral curves and exploring their connections to intersection theory on moduli spaces.

Contribution

It constructs a broad class of quantum Airy structures and extends topological recursion to singular spectral curves, revealing new links to intersection theory.

Findings

Constructed a large class of quantum Airy structures.

Extended topological recursion to singular spectral curves.

Connected topological recursion with intersection theory on moduli spaces.

Abstract

We give elements towards the classification of quantum Airy structures based on the $W(\mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $\mathfrak{gl}_r$ for twists by arbitrary elements of the Weyl group $\mathfrak{S}_{r}$. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion \`a la Chekhov-Eynard-Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard-Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves, giving a general ELSV-type representation for the topological recursion amplitudes on smooth curves, and formulate precise conjectures for application in open $r$-spin intersection theory.