Higher Airy structures, W algebras and topological recursion

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 $\mathfrak{g}={\mathfrak{gl}}_{N+1}$, ${\mathfrak{so}}_{2N}$ 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.

93 pages, v4: references added, many typos corrected

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• On higher Brézin-Gross-Witten tau-functions
• KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions