Simple Maps, Hurwitz Numbers, and Topological Recursion

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 $$x \leftrightarrow y$$ 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 $$(\lambda _1,\ldots ,\lambda _n)$$ ( λ 1 , … , λ n ) are strictly monotone double Hurwitz numbers with ramifications $$\lambda $$ λ above $$\infty $$ ∞ and $$(2,\ldots ,2)$$ ( 2 , … , 2 ) above 0. Combining with a recent result of Dubrovin et al. [24], this implies an ELSV-like formula for these Hurwitz numbers.

On the symplectically invariant variational principle and generating functions

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.