The spectral curve of the Eynard-Orantin recursion via the Laplace transform

Abstract

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 construction of the spectral curve and the recursion kernel from the unstable geometries of the original counting problem. We examine this construction using four concrete examples: Grothendieck's dessins d'enfants (or higher-genus analogue of the Catalan numbers), the intersection numbers of tautological cotangent classes on the moduli stack of stable pointed curves, single Hurwitz numbers, and the stationary Gromov-Witten invariants of the complex projective line.

49 pages, 7 figures

References in corpus (13)

• Open string amplitudes and large order behavior in topological string theory
• Torus knots and mirror symmetry
• 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
• Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
• Local Mirror Symmetry for One-Legged Topological Vertex
• Two Dimensional Kodaira-Spencer Theory and Three Dimensional Chern-Simons Gravity
• Hurwitz numbers, matrix models and enumerative geometry
• Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F_g invariants
• On computations of Hurwitz-Hodge integrals
• An algebra of power series arising in the intersection theory of moduli spaces of curves and in the enumeration of ramified coverings of the sphere
• On a proof of the Bouchard-Sulkowski conjecture

Cited by in corpus (22)

• The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures
• Quantum curves for Hitchin fibrations and the Eynard-Orantin theory
• Dubrovin's superpotential as a global spectral curve
• Quantum spectral curve for the Gromov-Witten theory of the complex projective line
• Equivalence of ELSV and Bouchard-Mariño conjectures for $r$-spin Hurwitz numbers
• A short overview of the "Topological recursion"
• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• Quantum curves and conformal field theory
• The Eynard--Orantin recursion for the total ancestor potential
• Double Hurwitz numbers: polynomiality, topological recursion and intersection theory
• Enumeration of chord diagrams on many intervals and their non-orientable analogs
• Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck's dessins d'enfants
• Combinatorics of Bousquet-Mélou-Schaeffer numbers in the light of topological recursion
• Matrix Field Theory
• Partial chord diagrams and matrix models
• Singular Values of Products of Ginibre Random Matrices
• A formalism of abstract quantum field theory of summation of fat graphs
• Feynman diagrams, ribbon graphs, and topological recursion of Eynard-Orantin
• Topological 1D Gravity, KP Hierarchy, and Orbifold Euler Characteristics of ${\overline{\mathcal{M}}}_{g,n}$
• Local topological recursion governs the enumeration of lattice points in ${\overline{\mathcal{M}}}_{g,n}$
• Topological Recursion for Generalized $bc$-Motzkin Numbers
• Quantum curve and bilinear Fermionic form for the orbifold Gromov-Witten theory of $\mathbb{P}[r]$