Higher Airy structures and topological recursion for singular spectral curves

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 à 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.

99 pages, 2 figures. v2.: added complete classification for genus zero invariants, Theorem 2.13, and more on intersection numbers in the regular case, sections 7.2.4, 7.5.4, & 7.5.5. Also numerous small corrections. v3.: Small corrections, most notably in section 8. Accepted to Annales de l'Institut Henri Poincaré D

References in corpus (3)

• Topological expansion of the Bethe ansatz, and quantum algebraic geometry
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• Open r-spin theory I: Foundations

Cited by in corpus (8)

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• KP integrability through the $x-y$ swap relation
• Taking limits in topological recursion
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• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• Topological recursion on transalgebraic spectral curves and Atlantes Hurwitz numbers
• A New Class of Higher Quantum Airy Structures as Modules of $\mathcal{W}({\mathfrak{gl}}_r)$-Algebras
• $\mathfrak{b}$-Hurwitz numbers from refined topological recursion