Simple maps, topological recursion and Hurwitz numbers
We call ordinary maps a certain type of graphs embedded on surfaces, in contrast to fully simple maps, which we introduce as maps with non-intersecting disjoint boundaries. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). We propose a combinatorial interpretation of the important and still mysterious symplectic transformation which exchanges $x$ and $y$ in the initial data of the TR (the spectral curve). We give elegant formulas for the disk and cylinder topologies which recover relations already known in the context of free probability. For genus zero we provide an enumerative geometric interpretation of the so-called higher order free cumulants, which suggests the possibility of a general theory of approximate higher order free cumulants taking into account the higher genus amplitudes. We provide a universal relation between possibly disconnected fully simple and ordinary maps through double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics. As a consequence, we obtain an ELSV-like formula for $2$-orbifold double strictly monotone Hurwitz numbers.
Elba García Failde, IPhT, Saclay