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Bertrand Eynard

Publications and source records attributed to Bertrand Eynard.

At least 19 recordsLinked to original sources
arXiv2026

Enumeration of plane hypermaps with a mixed boundary I

Plane hypermaps are plane maps endowed with a proper coloration of their inner faces in black or white. We consider the problem of enumerating plane hypermaps with prescribed face degrees and a $k$-alternating boundary condition: by this we mean the colors of inner faces incident to the outer face alternates at most $2k$ times when turning around the hypermap. The present paper deals with the cases $k=1,2$, the general case being left to the forthcoming part II. Our approach relies on the so-called slice decomposition and uses crucially the notion of accessibility, which exploits the canonical orientation of hypermaps and the marking variable $t$ associated with vertices, to enumerate pointed hypermaps by decomposing them according to the set of all vertices that can access to the marked vertex. This process enables us to express the generating functions of hypermaps with mixed boundaries in terms of the generating functions of hypermap slices and to recover, in a purely combinatorial way, some formulas previously obtained through algebraic methods.

math.CO↗
arXiv2026

Enumeration of general planar hypermaps with an alternating boundary

In this paper, we extend the enumerative study of planar hypermaps with an alternating boundary introduced in an earlier work of Bouttier and the second author. In that article, an explicit rational parametrization was obtained for the associated generating function in the case of m-constellations, using a variant of the kernel method. We develop here a new strategy to obtain an algebraic equation in the general case, which includes maps decorated by the Ising model, through a classical many-to-one correspondence. One of the main steps of our strategy is the simultaneous elimination of two catalytic variables. We then apply this strategy to the case of Ising quadrangulations, where we obtain an explicit rational parametrization. As a consequence, we show that some notable properties of the constellations case are no longer satisfied in general.

math.CO↗
arXiv2024

Existence of Boutroux curves, $g$-functions and spectral networks from Newton's polygon

We prove the existence of an algebraic plane curve of equation $P(x,y)=0$, with prescribed asymptotic behaviors at punctures, and with the Boutroux property, namely, periods have vanishing real part, i.e, $\Re(\int_\gamma y dx)=0$ for every closed loop $\gamma$. This has applications in the Riemann-Hilbert problem, in random matrix theory, in spectral networks, in WKB analysis and Stokes phenomenon, in algebraic and enumerative geometry, and many applications in mathematical physics. From Newton's polygon we can define an affine space such that there exists always a Boutroux curve. This result is applied to random matrix and asymptotic theory, in which a key ingredient is called the $g$-function, the function $g(x)=\int_o^x Y dX$ is a $g$-function precisely if and only if the algebraic plane curve is a Boutroux curve.

math-ph↗
arXiv2024

The factorial growth of topological recursion

We show that the $n$-point, genus-$g$ correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like $(2g - 2 + n)!$ as $g \rightarrow \infty$, which is the expected growth rate. This provides, in particular, an upper bound for many curve counting problems in large genus and serves as a preliminary step for a resurgence analysis.

math-ph↗
arXiv2024

Lecture on the combinatorial algebraic method for computing algebraic integrals

Consider an algebraic equation $P(x,y)=0$ where $P\in \mathbb C[x,y] $ (or $\mathbb F[x,y]$ with $\mathbb F\subset \mathbb C$ a subfield) is a bivariate polynomial, it defines a plane algebraic curve. We provide an efficient method for computing integrals of the type $ \int_γR(x,y)dx $ where $R(x,y)\in \mathbb C(x,y) $ is any rational fraction, and $y$ is solution of $P(x,y)=0$, and $γ$ any Jordan arc open or closed on the plane algebraic curve. The method uses only algebraic and combinatorial manipulations, it rests on the combinatorics of the Newton's polygon. We illustrate it with many practical examples.

math-ph↗
arXiv2024

Quantization of classical spectral curves via topological recursion

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

math-ph↗
arXiv2024

Hirota, Fay and Geometry

This is a review of the relationship between Fay identities and Hirota equations in integrable systems, reformulated in a geometric language compatible with recent Topological Recursion formalism. We write Hirota equations as trans-series, and Fay identities as spinor functional relations. We also recall several constructions of how some solutions to Fay/Hirota equations can be built from Riemann surface geometry.

math-ph↗
arXiv2023

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.

math-ph↗
arXiv2023

A natural basis for intersection numbers

We advertise elementary symmetric polynomials $e_i$ as the natural basis for generating series $A_{g,n}$ of intersection numbers of genus g and n marked points. Closed formulae for $A_{g,n}$ are known for genera $0$ and $1$ -- this approach provides formulae for $g = 2,3,4$, together with an algorithm to compute the formula for any g. The claimed naturality of the e_i basis relies in the unexpected vanishing of some coefficients with a clear pattern: we conjecture that $A_{g,n}$ can have at most $g$ factors $e_i$, with $i>1$, in its expansion. This observation promotes a paradigm for more general cohomology classes. As an application of the conjecture, we find new integral representations of $A_{g,n}$, which recover expressions for the Weil-Petersson volumes in terms of Bessel functions.

math.AG↗
arXiv2023

Resurgent Asymptotics of Jackiw-Teitelboim Gravity and the Nonperturbative Topological Recursion

Jackiw-Teitelboim dilaton-quantum-gravity localizes on a double-scaled random-matrix model, whose perturbative free energy is an asymptotic series. Understanding the resurgent properties of this asymptotic series, including its completion into a full transseries, requires understanding the nonperturbative instanton sectors of the matrix model for Jackiw-Teitelboim gravity. The present work addresses this question by setting-up instanton calculus associated to eigenvalue tunneling (or ZZ-brane contributions), directly in the matrix model. In order to systematize such calculations, a nonperturbative extension of the topological recursion formalism is required -- which is herein both constructed and applied to the present problem. Large-order tests of the perturbative genus expansion validate the resurgent nature of Jackiw-Teitelboim gravity, both for its free energy and for its (multi-resolvent) correlation functions. Both ZZ and FZZT nonperturbative effects are required by resurgence, and they further display resonance upon the Borel plane. Finally, the resurgence properties of the multi-resolvent correlation functions yield new and improved resurgence formulae for the large-genus growth of Weil-Petersson volumes.

hep-th↗
arXiv2023

Counting mobiles by integrable systems

Mobiles are a particular class of decorated plane trees which serve as codings for planar maps. Here we address the question of enumerating mobiles in their most general flavor, in correspondence with planar Eulerian (i.e., bicolored) maps. We show that the generating functions for such mobiles satisfy a number of recursive equations which lie in the field of integrable systems, leading us to explicit expressions for these generating functions as ratios of particular determinants. In particular we recover known results for mobiles associated with uncolored maps and prove some conjectured formulas for the generating functions of mobiles associated with $p$-constellations.

math-ph↗
arXiv2023

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich graphs and find combinatorial relations among them. We call the main type ciliated maps and use the auxiliary ones to show they satisfy a Tutte recursion that we turn into a combinatorial interpretation of the loop equations of topological recursion for a large class of spectral curves. It follows that ciliated maps, which are Feynman graphs for the Generalised Kontsevich matrix Model (GKM), are computed by topological recursion. Our particular instance of the GKM relates to the r-KdV integrable hierarchy and since the string solution of the latter encodes intersection numbers with Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin intersection numbers, implying that they are also governed by topological recursion. In turn, this paves the way towards a combinatorial understanding of Witten's class. This new topological recursion perspective on the GKM provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves.

math.CO↗
arXiv2023

Resurgent large genus asymptotics of intersection numbers

In this paper, we present a novel approach for computing the large genus asymptotics of intersection numbers. Our strategy is based on a resurgent analysis of the $n$-point functions of such intersection numbers, which are computed via determinantal formulae, and relies on the presence of a quantum curve. With this approach, we are able to extend the recent results of Aggarwal for Witten-Kontsevich intersection numbers with the computation of all subleading corrections, proving a conjecture of Guo-Yang, and to obtain new results on $r$-spin and Theta-class intersection numbers.

math.AG↗
arXiv2023

Recursions and ODEs for the correlators in integrable systems and random matrices

An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $\Psi$ living in a Lie group $G$, which satisfies some differential equations with rational coefficients. From this wave function, it is usual to define a sequence of ``correlators" $W_n$, that play an important role in many applications in mathematical physics. Here, we show how to systematically obtain ordinary differential equations (ODE) and recursion relations with polynomial coefficients for the correlators. An application is random matrix theory, where the wave functions are the expectation value of the characteristic polynomial, they form a family of orthogonal polynomials, and are known to satisfy an integrable system. The correlators are then the correlation functions of resolvents or of eigenvalue densities. We give the ODE and recursion on the matrix size that they satisfy. In addition, we discuss generic Fuchsian systems, namely, Schlesinger systems.

math-ph↗
arXiv2023

A new formula for intersection numbers

We propose a new formula to compute Witten--Kontsevich intersection numbers. It is a closed formula, not involving recursion neither solving equations. It only involves sums over partitions of products of factorials, double factorials and Kostka numbers (numbers of semi-standard tableau of given shape and weight) with bounded weights. As an application, we prove a conjecture of [ELO21] stating that the generating polynomials of the intersection numbers expressed in the basis of elementary symmetric polynomials have an unexpected vanishing of their coefficients.

math-ph↗
arXiv2020

From the quantum geometry of Fuchsian systems to conformal blocks of W-algebras

We consider the moduli space of holomorphic principal bundles for reductive Lie groups over Riemann surfaces (possibly with boundaries) and equipped with meromorphic connections. We associate to this space a point-wise notion of quantum spectral curve whose generalized periods define a new set of moduli. We define homology cycles and differential forms of the quantum spectral curve, allowing to derive quantum analogs of the form-cycle duality and Riemann bilinear identities of classical geometry. A tau-function is introduced for this system in the form of a theta-series and in such a way that the variations of its coefficients with respect to moduli, isomonodromic or not, can be computed as quantum period integrals. This lays new grounds to relate our study to that of integrable hierarchies, isomonodromic deformation of meromorphic connections and non-perturbative topological string theory. In turn, we define amplitudes on the quantum spectral curve which have an interpretation in conformal field theory when the Lie algebra is assumed to be simply-laced: they coincide with correlation functions involving twisted chiral fields of an affine Lie algebra at level one. The singularities at the punctures are interpreted as primary fields of the associated Casimir W-algebra. The amplitudes are moreover related by W-constraints, so-called loop equations, allowing one to compute recursively a certain asymptotic expansion of the tau-function, namely the one corresponding both to the heavy-charge regime of conformal field theory and to the weak-coupling regime of topological string theory.

math-ph↗
arXiv2019

The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition

In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes", all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is a degree 2 second-kind cycle, a Sato shift is an addition of a 3rd kind cycle, the Hirota equations amount to saying that merging 3rd kind cycles (monopoles) yields a 2nd kind cycle (dipole). The lecture is divided into 3 parts: 1) classical case, perturbative: the spectral curve is a ramified cover of a base Riemann surface -- with some additional structure -- and the integrable system is defined as a formal power series of a small "dispersion" parameter $ε$. 2) dispersive classical case, non perturbative: the spectral curve is defined not as a ramified cover (which would be a bundle with discrete fiber), but as a vector bundle -- whose dispersionless limit consists in chosing a finite set of vectors in each fiber. 3) non-commutative case, and perturbative. The spectral curve is here a "non-commutative" surface, whose geometry will be defined in lecture III. 4) the full non-commutative dispersionless theory is under development is not presented in these lectures.

math-ph↗