Topological recursion for fully simple maps from ciliated maps

Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve $(y, x)$, making use of the topological recursion for ciliated maps arXiv:2105.08035.

Relating ordinary and fully simple maps via monotone Hurwitz numbers

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.

Geometric recursion

We propose a general theory to construct functorial assignments $Sigma longmapsto Omega_{Sigma} in E(Sigma)$ for a large class of functors $E$ from a certain category of bordered surfaces to a suitable target category of topological vector spaces. The construction proceeds by successive excisions of homotopy classes of embedded pairs of pants, and thus by induction on the Euler characteristic. We provide sufficient conditions to guarantee the infinite sums appearing in this construction converge. In particular, we can generate mapping class group invariant vectors $Omega_{Sigma} in E(Sigma)$. The initial data for the recursion encode the cases when $Sigma$ is a pair of pants or a torus with one boundary, as well as the recursion kernels used for glueing. We give this construction the name of Geometric Recursion (GR). As a first application, we demonstrate that our formalism produce a large class of measurable functions on the moduli space of bordered Riemann surfaces. Under certain conditions, the functions produced by the geometric recursion can be integrated with respect to the Weil--Petersson measure on moduli spaces with fixed boundary lengths, and we show that the integrals satisfy a topological recursion (TR) generalizing the one of Eynard and Orantin. We establish a generalization of Mirzakhani--McShane identities, namely that multiplicative statistics of hyperbolic lengths of multicurves can be computed by GR, and thus their integrals satisfy TR. As a corollary, we find an interpretation of the intersection indices of the Chern character of bundles of conformal blocks in terms of the aforementioned statistics. The theory has however a wider scope than functions on Teichmuller space, which will be explored in subsequent papers; one expects that many functorial objects in low-dimensional geometry could be constructed by variants of our new geometric recursion.

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 free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x leftrightarrow y$ 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 symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. 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 $(lambda_1,ldots,lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $lambda$ above $infty$ and $(2,ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.

Nesting statistics in the $O(n)$ loop model on random maps of arbitrary topologies

We pursue the analysis of nesting statistics in the $O(n)$ loop model on random maps, initiated for maps with the topology of disks and cylinders in math-ph/1605.02239, here for arbitrary topologies. For this purpose we rely on the topological recursion results of math-ph/0910.5896 and math-ph/1303.5808 for the enumeration of maps in the $O(n)$ model. We characterize the generating series of maps of genus $g$ with $k$ marked points and $k$ boundaries and realizing a fixed nesting graph. These generating series are amenable to explicit computations in the loop model with bending energy on triangulations, and we characterize their behavior at criticality in the dense and in the dilute phase.

Modular functors, cohomological field theories and topological recursion

Given a topological modular functor $mathcal{V}$ in the sense of Walker cite{Walker}, we construct vector bundles over $bar{mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction depends on a determination of the logarithm of the eigenvalues of the Dehn twist and central element actions. We show that the intersection of the Chern class with the $psi$-classes in $bar{mathcal{M}}_{g,n}$ is computed by the topological recursion of cite{EOFg}, for a local spectral curve that we describe. In particular, we show how the Verlinde formula for the dimensions $D_{vec{lambda}}(mathbf{Sigma}_{g,n}) = dim mathcal{V}_{vec{lambda}}(mathbf{Sigma}_{g,n})$ is retrieved from the topological recursion. We analyze the consequences of our result on two examples: modular functors associated to a finite group $G$ (for which $D_{vec{lambda}}(mathbf{Sigma}_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $vec{lambda}$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $mathcal{V}_{vec{lambda}}(mathbf{Sigma}_{g,n})$ is the Verlinde bundle).

Abstract loop equations, topological recursion, and applications

We formulate a notion of abstract loop equations, and show that their solution is provided by a topological recursion under some assumptions, in particular the result takes a universal form. The Schwinger-Dyson equation of the one and two hermitian matrix models, and of the O(n) model appear as special cases. We study applications to repulsive particles systems, and explain how our notion of loop equations are related to Virasoro constraints. Then, as a special case, we study in detail applications to enumeration problems in a general class of non-intersecting loop models on the random lattice of all topologies, to SU(N) Chern-Simons invariants of torus knots in the large N expansion. We also mention an application to Liouville theory on surfaces of positive genus.