# Topological Recursion and Combinatorics

## Friday, November 5, 2021

Topological recursion is a method of finding formulas for an infinite sequence of series or n-forms by means of describing them in a recursive way in terms of genus and boundary points of certain topological surfaces.
While topological recursion was originally discovered in Random Matrix Theory, and could be traced back to the Harer-Zagier formula, it was until Chekhov, Eynard and Orantin (2007) that is was systematically studied. Since then it has found applications in different areas in mathematics and physics such as enumerative geometry, volumes of moduli spaces, Gromov-Witten invariants, integrable systems, geometric quantization, mirror symmetry, matrix models, knot theory and string theory.

This 1/2-day series of seminar talks aims at exploring combinatorial aspects relevant to the theory of topological recursion in Random Matrix Models and to widen the bridge to free probability and its generalizations such as higher order freeness or infinitesimal freeness more transparent.

Sessions will be hosted via Zoom and/or gather.town.

Registration here. Please send us an email if you want to join last-minute.

# Titles and Abstracts

**Title:** Topological recursion, fully simple maps and Hurwitz numbers

**Abstract:** 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. I will introduce the combinatorial models and the corresponding matrix models. It is well-known that the generating series of ordinary maps satisfy a universal recursive procedure, called topological recursion (TR). I will introduce TR and give a combinatorial interpretation, through fully simple maps, of the important and still mysterious symplectic transformation which exchanges x and y in the initial data (spectral curve) of the TR. Using combinatorial decompositions of maps, we find closed formulas for the disk and cylinder topologies, which recover relations already known in the context of free probability. Finally, we present a universal relation between fully simple and ordinary observables involving double monotone Hurwitz numbers, which can be proved either using matrix models or bijective combinatorics. With these relations we pave the way for Séverin Charbonnier, who will elaborate on the connection of this duality that arises in the TR world to the duality between moments and free cumulants in free probability. This talk is based on joint work with subsets of G. Borot, S. Charbonnier and N. Do.

**Title:** Functional relations for generating functions of higher order moments and cumulants in free probability

**Abstract:** Free moments and free cumulants were introduced in order to study a notion of independence of variables in non-commutative probability spaces. Voiculescu proved that the generating functions of free moments and cumulants satisfy a functional relation, called the R-transform formula. Later, in 2007, Collins, Mingo, Śniady and Speicher defined higher order freeness and derived the functional relation for the second order. In collaboration with Gaëtan Borot, Elba Garcia-Failde, Felix Leid and Sergey Shadrin, we prove a general functional relation for any order.
I will first expose various concepts of free probabilities that we are interested in, such as free moments, cumulants, and their higher order counterparts. I will give the connection between those objects, and the combinatorial setup introduced in Elba Garcia-Failde's talk, in particular the generating functions of (fully simple) maps and the R-transform formulas of first and second order. Second, using a result of Elba Garcia-Failde's talk, I will give a general version of the functional relation for higher orders.

**Title:** Infinitesimal freeness and a connection to higher order freeness

**Abstract:** Infinitesimal freeness is a special case of type B freeness and as such has much of the combinatorial machinery of free probability available. I will review the basic points of infinitesimal freeness and some unitarily invariant random matrix examples of asymptotic infinitesimal freeness. In the case of orthogonally invariant ensembles something a little different emerges. I will present some examples that indicate a connection to higher order freeness.

**Title:** Oscillatory asymptotic for HCIZ and BGW integrals

**Abstract:** The HCIZ and BGW integrals are a pair of oscillatory matrix integrals which arise as analytic continuations of the Fourier transform of Haar measure on a given coadjoint orbit of U(N), or of Haar measure on U(N) itself. While physicists believe that these integrals have large N topological expansions, both the existence and nature of these expansions has remained mathematically uncertain. I will explain that the topological expansions of the HCIZ and BGW integrals do indeed exist in the oscillatory regime at sufficiently strong coupling, and that they are generating functions for the monotone double and single Hurwitz numbers, respectively.