• Corpus ID: 26575284

A short overview of the "Topological recursion"

  title={A short overview of the "Topological recursion"},
  author={Bertrand Eynard},
  journal={arXiv: Mathematical Physics},
  • B. Eynard
  • Published 10 December 2014
  • Mathematics
  • arXiv: Mathematical Physics
This is the long version of the ICM2014 proceedings. It consists in a short overview of the ”topological recursion”, a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method recover many classical invariants, like Gromov–Witten invariants, or knot polynomials (Jones, HOMFLY… 

Figures from this paper

Quantum curves from refined topological recursion: the genus 0 case
. We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing
Lectures on the Topological Recursion for Higgs Bundles and Quantum Curves
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the
on the Topological Recursion for Higgs Bundles and Quantum Curves Permalink
  • Mathematics
  • 2015
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the
The ABCD of topological recursion
Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector
Topological recursion
In this snapshot we present the concept of topological recursion – a new, surprisingly powerful formalism at the border of mathematics and physics, which has been actively developed within the last
Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson–Pandharipande–Tseng formula
For orbifold Hurwitz numbers, a new proof of the spectral curve topological recursion is given in the sense of Chekhov, Eynard and Orantin, where the main new step compared to the existing proofs is a direct combinatorial proof of their quasi-polynomiality.
Enumeration of non-oriented maps via integrability
ABSTRACT. In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence
Topological Recursion, Airy structures in the space of cycles
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
Analyticity of the free energy for quantum Airy structures
  • B. Ruba
  • Mathematics
    Journal of Physics A: Mathematical and Theoretical
  • 2020
It is shown that the free energy associated to a finite-dimensional Airy structure is an analytic function at each finite order of the -expansion. Its terms are interpreted as objects living on the
An invitation to 2D TQFT and quantization of Hitchin spectral curves
This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor


CFT and topological recursion
We study the quasiclassical expansion associated with a complex curve. In a more specific context this is the 1/N expansion in U(N)-invariant matrix integrals. We compare two approaches, the CFT
New recursive residue formulas for the topological expansion of the Cauchy Matrix Model
In a recent work [1] we consider the topological expansion for the non-mixed observables (including the free energy) for the formal Cauchy matrix model. The only restriction in [1] was the fact that
Algebraic methods in random matrices and enumerative geometry
We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one
All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials
We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our
Hurwitz numbers, matrix models and enumerative geometry
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric
A Generalized Topological Recursion for Arbitrary Ramification
The Eynard–Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple
Invariants of algebraic curves and topological expansion
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties.
The spectral curve of the Eynard-Orantin recursion via the Laplace transform
The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform
Geometry of Spectral Curves and All Order Dispersive Integrable System
We propose a definition for a Tau function and a spinor kernel (closely related to Baker{Akhiezer functions), where times parametrize slow (of order 1=N) deformations of an algebraic plane curve.
Free energy topological expansion for the 2-matrix model
We compute the complete topological expansion of the formal hermitian two-matrix model. For this, we refine the previously formulated diagrammatic rules for computing the 1/N expansion of the