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Simple maps, Hurwitz numbers, and topological recursion. (English) Zbl 1457.37055

If a connected graph \(\Gamma\) is embedded into a connected orientable surface \(X\) of genus \(g\) such that \(X\backslash\Gamma\) is a disjoint union of connected components, called faces, each of them homeomorphic to an open disk, then \(\Gamma\) is called an embedded graph of genus \(g\). The length of a face is the the number of edges belonging to it. An embedded graph is said to have \(n\) boundaries, when it has \(n\) marked faces, labeled \(1,\dots,n\), containing a marked edge, called root, represented by an arrow following the convention that the marked face sits on the left side of the root. A face which is not marked is called an inner face. Two embedded graphs \(\Gamma_i\subset X_i\), \(i=1,2\), are isomorphic if there exists an orientation preserving homeomorphism \(\varphi: X_1\to X_2\) such that \(\varphi_{|\Gamma_1}\) is a graph isomorphism between \(\Gamma_1\) and \(\Gamma_2\), and the restriction of \(\varphi\) to the marked edges is the identity. An isomorphism class of embedded graphs is called a map. Maps can be viewed as surfaces obtained by gluing polygons. A map of genus \(g=0\) is called planar. A map of genus \(g\) with \(n\) boundaries is called map of topology \((g,n)\). For the cases \((0,1)\) and \((0,2)\), maps are called disks and cylinders, respectively. If \(\mathbb{M}^{[g]}_n\) is the set of maps of genus \(g\) with \(n\) boundaries, then the generating series of maps of genus \(g\) and \(n\) boundaries of respective lengths \(l_1,\dots,l_n\) is defined as \[F^{[g]}_{l_1,\dots,l_n} =\sum\limits_{\mathcal{M}\in\mathbb{M}^{[g]}_n} \frac{\Pi_{j\ge 1}t^{n_j(\mathcal{M})}_j}{|\mathrm{Aut}\mathcal{M}|} \prod\limits^n_{i=1}\delta_{l_i,\ell_i(\mathcal{M})},\] where \(n_j (\mathcal{M})\) denotes the number of unmarked faces of length \(j\), and \(\ell_i(\mathcal{M})\) denotes the length of the \(i\)-th boundary of \(\mathcal{M}\). A boundary \(B\) is called simple if no more than two edges belonging to \(B\) are incident to a vertex, and a map is called simple if all boundaries are simple.
In this paper, the authors introduce the notion of fully simple maps. A boundary \(B\) is called fully simple if no more than two edges belonging to any boundary are incident to a vertex of \(B\), and a map is said to be fully simple if all boundaries are fully simple. A fully simple boundary can be visualized as a simple boundary which moreover does not share any vertex with any other boundary. The authors study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. They show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks, and they also obtain a formula for cylinders. The authors conjecture that the generating series of fully simple maps are computed by the topological recursion after the exchange of \(x\) and \(y\), and they propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the \(1\)-Hermitian matrix model. This conjecture is considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion. Finally, they deduce the universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers.

MSC:

37E15 Combinatorial dynamics (types of periodic orbits)
05C30 Enumeration in graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
53D42 Symplectic field theory; contact homology
15B52 Random matrices (algebraic aspects)
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
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