Communications in Number Theory and Physics

Volume 8 (2014)

Number 4

Quantum curves for the enumeration of ribbon graphs and hypermaps

Pages: 677 – 701



Norman Do (School of Mathematical Sciences, Monash University, Victoria, Australia)

David Manescu (School of Mathematics and Statistics, The University of Sydney, Australia)


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 quantization 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 generalizations 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.

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