This is a joint work with Vincent Bouchard, Dmitry Noshchenko and Nitin Chidambaram.

Quantum Airy structure were introduced by Kontsevich and Soibelman, and essentially formalize the notion of "algebra of differential constraints" in such a way that existence and uniqueness of a function annihilated by these constraints is guaranteed. We show that such constraints can be produced from certain modules over W-algebras, and for type A_r we show that these constraints are equivalent to computing the partition function via the higher topological recursion of Bouchard and Eynard. This applies for instance to the constraints for the total descendent potential of the A_{r}-singularity (intersection numbers of the Witten r-spin class), to the constraints for the open intersection numbers. We also get other examples of quantum Airy structures, for which the generating series are expected to have an enumerative geometry meaning (e.g. open r-spin intersection numbers ?) and integrability properties.

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