Gromov-Witten theory, Hurwitz numbers, and Matrix models, I

Abstract

The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.

Latex, 107 pages, 9 figures, updated version

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• Virasoro constraints and the Chern classes of the Hodge bundle
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Cited by in corpus (75)

• An algebro-geometric proof of Witten's conjecture
• Hodge integrals and invariants of the unknot
• Algebra of differential operators associated with Young diagrams
• Three questions in Gromov-Witten theory
• On Combinatorial Expansions of Conformal Blocks
• Algebraic methods in random matrices and enumerative geometry
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• Matrix Models for Random Partitions
• Phase transitions, double-scaling limit, and topological strings
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• The equivariant Gromov-Witten theory of P^1
• Topological strings and large N phase transitions II: Chiral expansion of q-deformed Yang-Mills theory
• Orbifold Hurwitz numbers and Eynard-Orantin invariants
• Localization in Gromov-Witten Theory and Orbifold Gromov-Witten Theory
• Intersection theory from duality and replica
• The spectral curve of the Eynard-Orantin recursion via the Laplace transform
• Polynomiality of Hurwitz numbers, Bouchard-Mariño conjecture, and a new proof of the ELSV formula
• Hurwitz numbers, matrix models and enumerative geometry
• A simple proof of Witten conjecture through localization
• Matrix models and a proof of the open analog of Witten's conjecture
• Geometry of meromorphic functions and intersections on moduli spaces of curves
• Holomorphic maps and the complete 1/N expansion of 2D SU(N) Yang-Mills
• On a worldsheet dual of the Gaussian matrix model
• The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers
• KP hierarchy for Hodge integrals
• Tautological and non-tautological cohomology of the moduli space of curves
• The uses of random partitions
• Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
• Random surfaces enumerating algebraic curves
• The moduli space of curves, double Hurwitz numbers, and Faber's intersection number conjecture
• Witten's conjecture, Virasoro conjecture, and invariance of tautological equations
• Monotone Hurwitz numbers and the HCIZ integral II
• Generating functions for intersection numbers on moduli spaces of curves
• Nonlinear PDEs for Fredholm determinants arising from string equations
• Generalized string equations for double Hurwitz numbers
• Kontsevich-Witten Model From 2+1 Gravity: New Exact Combinatorial Solution
• Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes
• Intersection numbers with Witten's top Chern class
• JT gravity and the asymptotic Weil-Petersson volume
• Tautological relations in Hodge field theory
• Quantization of Harer-Zagier formulas
• Formulae of one-partition and two-partition Hodge integrals
• Combinatorial Floer Homology
• Towards large genus asymtotics of intersection numbers on moduli spaces of curves
• Topological String Correlators from Matrix Models
• Cohomology of moduli spaces
• Enumeration of ramified coverings of the sphere and 2-dimensional gravity
• The Quantum Orbifold Cohomology of Weighted Projective Spaces
• Descendent integrals and tautological rings of moduli spaces of curves
• Mirzakharni's recursion formula is equivalent to the Witten-Kontsevich theorem
• A generating function of the number of homomorphisms from a surface group into a finite group
• Lectures and notes: Mirzakhani's volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers
• Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds
• Virasoro constraints for target curves
• The Toda conjecture
• Mirror Symmetry And Quantum Geometry
• Algebraic structures on the topology of moduli spaces of curves and maps
• Polynomial recursion formula for linear Hodge integrals
• Higher-genus Gromov-Witten invariants as genus 0 invariants of symmetric products
• Lectures on the ELSV formula
• Uniform Lower Bound for Intersection Numbers of $\psi$-Classes
• Abelian Hurwitz-Hodge integrals
• Intersection numbers and automorphisms of stable curves
• Tevelev degrees and Hurwitz moduli spaces
• Combinatorics of tropical Hurwitz cycles
• On the large genus asymptotics of psi-class intersection numbers
• On a Zeuthen-type problem
• Intersections in genus 3 and the Boussinesq hierarchy
• The Hodge-FVH Correspondence
• A connection between the Kontsevich-Witten and Brezin-Gross-Witten tau-functions
• Aspects géométriques et intégrables des modèles de matrices aléatoires
• Symplectic Geometry of Moduli Spaces of Hurwitz Covers
• Mirzakhani's recursion formula on Weil-Petersson volume and applications
• Gromov--Witten invariants of the Riemann sphere
• Gromov-Witten invariants of toric Calabi-Yau threefolds