A short overview of the "Topological recursion"

Abstract

This review is an extended version of the Seoul ICM 2014 proceedings.It is 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,...). In this short review, we give some examples, give definitions, and review some properties and applications of the formalism.

arXiv comment2nd version: misprints corrected, and small modification of the abstract

References in corpus (36)

• All genus correlation functions for the hermitian 1-matrix model
• Open string amplitudes and large order behavior in topological string theory
• Hermitean matrix model free energy: Feynman graph technique for all genera
• Free energy topological expansion for the 2-matrix model
• Holomorphic anomaly and matrix models
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• Solving Virasoro Constraints in Matrix Models
• Algebraic methods in random matrices and enumerative geometry
• Conformal Field Theory Techniques in Random Matrix models
• A holomorphic and background independent partition function for matrix models and topological strings
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• Lectures on complex geometry, Calabi-Yau manifolds and toric geometry
• Localization and Gluing of Topological Amplitudes
• Symplectic invariants, Virasoro constraints and Givental decomposition
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Intersection numbers of spectral curves
• Local Mirror Symmetry for One-Legged Topological Vertex
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• CFT and topological recursion
• Localization in Gromov-Witten Theory and Orbifold Gromov-Witten Theory
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Geometry of Spectral Curves and All Order Dispersive Integrable System
• Determinantal formulae and loop equations
• A generalized topological recursion for arbitrary ramification
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• Gromov-Witten invariants of $\bp^1$ and Eynard-Orantin invariants
• Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
• Bouchard-Klemm-Marino-Pasquetti Conjecture for ${\mathbb{C}}^3$
• All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds
• An introduction to the volume conjecture and its generalizations
• About the x-y symmetry of the F_g algebraic invariants
• The colored Jones polynomials and the simplicial volume of a knot
• On a Proof of a Conjecture of Marino-Vafa on Hodge Integrals
• On a proof of the Bouchard-Sulkowski conjecture

Cited by in corpus (18)

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• Riemann-Hilbert problems from Donaldson-Thomas theory
• Dubrovin's superpotential as a global spectral curve
• The ABCD of topological recursion
• JT gravity and the asymptotic Weil-Petersson volume
• From Noncommutative Geometry to Random Matrix Theory
• A Laplacian to compute intersection numbers on ${\bar{\mathcal{M}}}_{g,n}$ and correlation functions in NCQFT
• Path integrals in JT gravity and Virasoro constraints
• Correlation functions of $\mathrm{U}(N)$-tensor models and their Schwinger-Dyson equations
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Large genus behavior of topological recursion
• Topological Recursion, Airy structures in the space of cycles
• Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories
• Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin
• Topological recursion of Eynard-Orantin and the Harmonic Oscillator
• Enumeration of non-oriented maps via integrability
• Volume of the moduli space of unmarked bounded positive convex ${\mathbb{RP}}^2$ structures
• Metropolis updates for Diagrammatic Monte-Carlo algorithms from Schwinger-Dyson equations