Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Combinatorial interpretation of the symplectic invariance of topological recursion

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 (spectral curve) of the TR. We give closed 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 higher genus corrections. We also give a universal relation between fully simple and ordinary maps involving double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics. As a consequence, we obtain a new ELSV-like formula, which is a relation between Hurwitz theory and intersection numbers on the moduli spaces of curves.