Quantum curves for the enumeration of ribbon graphs and hypermaps

Abstract

The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-commutative quantisation whose associated differential operator annihilates the partition function for the spectral curve. In this paper, we determine the quantum curves and partition functions for an infinite sequence of enumerative problems involving generalisations of ribbon graphs known as hypermaps. These results give rise to an explicit conjecture relating hypermap enumeration to the topological recursion and we provide evidence to support this conjecture.

15 pages

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• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Quantum spectral curve for the Gromov-Witten theory of the complex projective line
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Cited by in corpus (3)

• Reconstructing WKB from topological recursion
• Topological recursion and a quantum curve for monotone Hurwitz numbers
• Integrable differential systems of topological type and reconstruction by the topological recursion