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.

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” Ψ 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” Wn, 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.

Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and integrable systems in the topological recursion approach. They parametrize deformations, implementing the special geometry, where moduli are periods, and derivatives with respect to moduli are other periods, or more generally "integrals", whence the name "generalized cycles". They appeared over the years in various works, each time in specific applied frameworks, and here we provide a comprehensive self-contained corpus of definitions and properties for a very general setting. The geometry of generalized cycles is also fascinating by itself.

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.

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.

We provide explicit expressions of ABCD tensors for the most classical classes of spectral curves. And we discuss algorithmic implementation of Topological Recursion.

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.

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 →∞, 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.

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 with 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.

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, (∫_γ y dx)=0 for every closed loop γ. 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)=∫_o^x Y dX is a g-function precisely if and only if the algebraic plane curve is a Boutroux curve.

Consider an algebraic equation P(x,y)=0 where P∈ℂ[x,y] (or 𝔽[x,y] with 𝔽⊂ℂ a subfield) is a bivariate polynomial, it defines a plane algebraic curve. We provide an efficient method for computing integrals of the type ∫_γ R(x,y)dx where R(x,y)∈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.

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 Painlev\'e 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

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.

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.

The zero locus of a bivariate polynomial P(x,y)=0 defines a compact Riemann surface \Sigma . The fundamental second kind differential is a symmetric 1\otimes 1 -form on \Sigma\times\Sigma that has a double pole at coinciding points and no other pole. As its name indicates, this is one of the most important geometric objects on a Riemann surface. Here we give a rational expression in terms of combinatorics of the Newton's polygon of P , involving only integer combinations of products of coefficients of P . Since the expression uses only combinatorics, the coefficients are in the same field as the coefficients of P .

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.

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.

Delaunay triangulations provide a bijection between a set of $N+3$ points in the complex plane, and the set of triangulations with given circumcircle intersection angles. The uniform Lebesgue measure on these angles translates into a K\"ahler measure for Delaunay triangulations, or equivalently on the moduli space $\mathcal M_{0,N+3}$ of genus zero Riemann surfaces with $N+3$ marked points. We study the properties of this measure. First we relate it to the topological Weil-Petersson symplectic form on the moduli space $\mathcal M_{0,N+3}$. Then we show that this measure, properly extended to the space of all triangulations on the plane, has maximality properties for Delaunay triangulations. Finally we show, using new local inequalities on the measures, that the volume $\mathcal{V}_N$ on triangulations with $N+3$ points is monotonically increasing when a point is added, $N\to N+1$. We expect that this can be a step towards seeing that the large $N$ limit of random triangulations can tend to the Liouville conformal field theory.

Topological recursion associates to a spectral curve, a sequence of meromorphic differential forms. A tangent space to the "moduli space" of spectral curves (its space of deformations) is locally described by meromorphic 1-forms, and we use form-cycle duality to re-express it in terms of cycles (generalized cycles). This formulation allows to express the ABCD tensors of Quantum Airy Structures acting on the vector space of cycles, in an intrinsic spectral-curve geometric way.

We show that for a rather generic set of regular spectral curves, the Topological-Recursion invariants F_g grow at most like O((β g)! r^-g) with some r>0 and β≤ 5.