Simple Maps, Hurwitz Numbers, and Topological Recursion
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Communications in
Mathematical Physics
Simple Maps, Hurwitz Numbers, and Topological Recursion Gaëtan Borot, Elba Garcia-Failde Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. E-mail: [email protected]; [email protected] Received: 20 December 2018 / Accepted: 5 June 2020 © The Author(s) 2020
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 (higher order) free cumulants established by Collins et al. [22], and implement the symplectic transformation x ↔ 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 the symplectic invariance for the 1-hermitian matrix model, which is believed to be true but has not been proved yet. Our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion. 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 (λ1 , . . . , λn ) are strictly monotone double Hurwitz numbers with ramifications λ above ∞ and (2, . . . , 2) above 0. Combining with a recent result of Dubrovin et al. [24], this implies an ELSV-like formula for these Hurwitz numbers.
We thank Roland Speicher for free discussions, Di Yang for explaining us the result of [24], Dominique Poulhalon and Guillaume Chapuy for bringing to our attention the literature on non-separable maps, Sergey Shadrin and Danilo Lewa´nski for useful discussions on references and clearer presentations of some results, and Max Karev for discussing references. This work benefited from the support of the Max-Planck Gesellschaft.
G. Borot, E. Garcia-Failde
Contents
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Disks and cylinders via combinatorics . . . . . . . . . . . . . . . . . 1.2 Matrix model interpretation and consequences . . . . . . . . . . . . . 1.3 Topological recursion interpretation . . . . . . . . . . . . . . . . . . 1.4 Application to free probability . . . . . . . . . . . . . . . . . . . . . 2. Objects of Study .
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