Higher Airy structures, W algebras and topological recursion

TL;DR

This paper introduces higher quantum Airy structures generalizing previous models by allowing differential operators of any order, explores their algebraic and geometric properties, and links them to topological recursion and intersection theory.

Contribution

It constructs new classes of higher quantum Airy structures from W algebras, characterizes spectral curves for topological recursion, and connects these structures to geometric intersection theory.

Findings

New higher quantum Airy structures from W algebras at self-dual level.

Characterization of spectral curves where topological recursion applies.

Equivalence of topological recursion to W constraints as higher quantum Airy structures.

Abstract

We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{g})$ algebras at self-dual level, with $\mathfrak{g}= \mathfrak{gl}_{N+1}$, $\mathfrak{so}_{2 N }$ or $\mathfrak{e}_N$. We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these $\mathcal{W}$ constraints have already appeared in the literature, but we find many new ones. For $\mathfrak{gl}_{N+1}$ our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard-Eynard topological recursion gives symmetric $\omega_{g,n}$s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to $\mathcal{W}(\mathfrak{gl})$ constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.