Simple maps, Hurwitz numbers, and Topological Recursion

Abstract

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x\leftrightarrow y$ on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of $x$ and $y$. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $({\lambda }_1,\dots ,{\lambda }_n)$ are strictly monotone double Hurwitz numbers with ramifications $\lambda$ above $\infty$ and $(2,\dots ,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.

66 pages, 7 figures

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Cited by in corpus (10)

• Explicit closed algebraic formulas for Orlov-Scherbin $n$-point functions
• Integer moments of complex Wishart matrices and Hurwitz numbers
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• From scalar fields on quantum spaces to blobbed topological recursion
• Perturbative and Geometric Analysis of the Quartic Kontsevich Model
• Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot--Garcia-Failde conjecture
• Relating ordinary and fully simple maps via monotone Hurwitz numbers
• Complete solution of the LSZ Model via Topological Recursion
• Symplectic duality for topological recursion
• Genus one free energy contribution to the quartic Kontsevich model