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Gaëtan Borot

Publications and source records attributed to Gaëtan Borot.

At least 37 records · Page 2Linked to original sources
arXiv2022

Around the combinatorial unit ball of measured foliations on bordered surfaces

The volume $\mathscr{B}_Σ^{\rm comb}(\mathbb{G})$ of the unit ball -- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of the space of measured foliations on a stable bordered surface $Σ$ appears as the prefactor of the polynomial growth of the number of multicurves on $Σ$. We find the range of $s \in \mathbb{R}$ for which $(\mathscr{B}_Σ^{\rm comb})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of $Σ$, in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently proved an optimal square-integrability.

math.GT↗
arXiv2021

Higher Airy structures, W algebras and topological recursion

We define higher quantum Airy structures as generalizations of the Kontsevich-Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of $\mathcal{W}(\mathfrak{g})$ algebras at self-dual level, with $\mathfrak{g}= \mathfrak{gl}_{N+1}$, $\mathfrak{so}_{2 N }$ or $\mathfrak{e}_N$. We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these $\mathcal{W}$ constraints have already appeared in the literature, but we find many new ones. For $\mathfrak{gl}_{N+1}$ our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard-Eynard topological recursion gives symmetric $ω_{g,n}$s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to $\mathcal{W}(\mathfrak{gl})$ constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

math-ph↗
arXiv2021

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.

math.CO↗
arXiv2021

On the Kontsevich geometry of the combinatorial Teichmüller space

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichmüller space $T_S^{comb}$ equipped with Kontsevich symplectic form $ω_K$, and then the usual Weil-Petersson geometry of Teichmüller space $T_S$. The basis for this is an identification of $T_S^{comb}$ with a space of measured foliations with transverse boundary conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen coordinates (defined similarly as Dehn-Thurston coordinates) and show they are Darboux for $ω_K$ (analog of Wolpert formula). We then set up the geometric recursion of Andersen-Borot-Orantin to produce mapping class group invariants functions on $T_S^{comb}$ whose integration with respect to Kontsevich volume form satisfy topological recursion. Further we establish an analog of Mirzakhani-McShane identities, and provide applications to the study of the enumeration of multicurves with respect to combinatorial lengths and Masur-Veech volumes. The formalism allows us to provide uniform and completely geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's topological recursion for lattice point count in the combinatorial moduli space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson volumes. We strengthen results of Mondello and Do on the convergence of hyperbolic geometry to combinatorial geometry along the rescaling flow, allowing us to flow systematically natural constructions on the usual Teichmüller space to their combinatorial analogue, such as a new derivation of the piecewise linear structure of $T_S^{comb}$ originally obtained in the work of Penner, as the limit under the flow of the smooth structure of $T_S$.

math.DG↗
arXiv2020

Topological recursion for Masur-Veech volumes

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

math.GT↗
arXiv2020

Double Hurwitz numbers: polynomiality, topological recursion and intersection theory

Double Hurwitz numbers enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification over two points and simple ramification elsewhere. In contrast to the single case, their underlying geometry is not well understood. In previous work by the second- and third-named authors, the double Hurwitz numbers were conjectured to satisfy a polynomiality structure and to be governed by the topological recursion, analogous to existing results concerning single Hurwitz numbers. In this paper, we resolve these conjectures by a careful analysis of the semi-infinite wedge representation for double Hurwitz numbers, by pushing further methods previously used for other Hurwitz problems. We deduce a preliminary version of an ELSV-like formula for double Hurwitz numbers, by deforming the Johnson-Pandharipande-Tseng formula for orbifold Hurwitz numbers and using properties of the topological recursion under variation of spectral curves. In the course of this analysis, we unveil certain vanishing properties of the Chiodo classes.

math.AG↗
arXiv2019

Loop equations for Gromov-Witten invariants of $\mathbb{P}^1$

We show that non-stationary Gromov-Witten invariants of $\mathbb{P}^1$ can be extracted from open periods of the Eynard-Orantin topological recursion correlators $ω_{g,n}$ whose Laurent series expansion at $\infty$ compute the stationary invariants. To do so, we overcome the technical difficulties to global loop equations for the spectral $x(z) = z + 1/z$ and $y(z) = \ln z$ from the local loop equations satisfied by the $ω_{g,n}$, and check these global loop equations are equivalent to the Virasoro constraints that are known to govern the full Gromov-Witten theory of $\mathbb{P}^1$.

math.AG↗
arXiv2019

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.

math.CO↗
arXiv2018

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 $(λ_1,\ldots,λ_n)$ are strictly monotone double Hurwitz numbers with ramifications $λ$ 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.

math-ph↗
arXiv2017

An introduction to random matrix theory

These are lectures notes for a 4h30 mini-course held in Ulaanbaatar, National University of Mongolia, August 5-7th 2015, at the summer school "Stochastic Processes and Applications". It aims at presenting an introduction to basic results of random matrix theory and some of its motivations, targeted to a large panel of students coming from statistics, finance, etc. Only a small background in probability is required.

math.PR↗
arXiv2017

Lecture notes on topological recursion and geometry

These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of topological recursion: in two dimensional quantum field theories, in cohomological field theories, in the computation of Weil-Petersson volumes of the moduli space of curves; (b) relate them more specifically to Eynard-Orantin topological recursion (revisited from Kontsevich-Soibelman point of view based on quantum Airy structures).

math-ph↗
arXiv2016

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 $ψ$-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 $D_{\vecλ}(\mathbfΣ_{g,n}) = \dim \mathcal{V}_{\vecλ}(\mathbfΣ_{g,n})$ 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 $D_{\vecλ}(\mathbfΣ_{g,n})$ enumerates certain $G$-principle bundles over a genus $g$ surface with $n$ boundary conditions specified by $\vecλ$), and the modular functor obtained from Wess-Zumino-Witten conformal field theory associated to a simple, simply-connected Lie group $G$ (for which $\mathcal{V}_{\vecλ}(\mathbfΣ_{g,n})$ is the Verlinde bundle).

math-ph↗
arXiv2015

Blobbed topological recursion: properties and applications

We study the set of solutions $(ω_{g,n})_{g \geq 0,n \geq 1}$ of abstract loop equations. We prove that $ω_{g,n}$ is determined by its purely holomorphic part: this results in a decomposition that we call "blobbed topological recursion". This is a generalization of the theory of the topological recursion, in which the initial data $(ω_{0,1},ω_{0,2})$ is enriched by non-zero symmetric holomorphic forms in $n$ variables $(ϕ_{g,n})_{2g - 2 + n > 0}$. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of $ω_{g,n}$ in terms of $ϕ_{g,n}$; (2) a graphical representation of $ω_{g,n}$ in terms of intersection numbers on the moduli space of curves; (3) variational formulae under infinitesimal transformation of $ϕ_{g,n}$ ; (4) a definition for the free energies $ω_{g,0} = F_g$ respecting the variational formulae. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

math-ph↗
arXiv2014

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

We study a class of scalar, linear, non-local Riemann-Hilbert problems (RHP) involving finite subgroups of PSL(2,C). We associate to such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. As an application, we study in detail the large N expansion of SU(N) or SO(N) or Sp(2N) Chern-Simons partition function Z_N(M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. It has a matrix model-like representation, whose spectral curve can be characterized in terms of a RHP as above. When pi_1(M) is finite (i.e. for manifolds M that are quotients of \mathbb{S}_{3} by a finite isometry group of type ADE), the Weyl group associated to the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large $N$ expansion of Z_N(M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in $M$.

math-ph↗
arXiv2013

Rational differential systems, loop equations, and application to the q-th reductions of KP

To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight hbar per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the q-th reductions of KP - which contain the (p,q) models as a specialization.

math-ph↗
arXiv2013

Formal multidimensional integrals, stuffed maps, and topological recursion

We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the topological recursion of [Eynard and Orantin, 2007] with initial conditions. In terms of a 1d gas of eigenvalues, this model includes - on top of the squared Vandermonde - multilinear interactions of any order between the eigenvalues. In this problem, the initial data (W10,W20) of the topological recursion is characterized: for W10, by a non-linear, non-local Riemann-Hilbert problem on a discontinuity locus to determine ; for W20, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - W10 being the generating series of disks and W20 that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps". In a sense, our results complete the program of the "moment method" initiated in the 90s to compute the formal 1/N in the one hermitian matrix model.

math-ph↗