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

TL;DR

This paper develops a uniform method to construct spectral curves and recursion kernels for the Eynard-Orantin recursion, connecting geometric enumeration problems with spectral curve analysis through Laplace transforms.

Contribution

It introduces a new uniform construction of spectral curves and kernels from unstable geometries, applicable to various geometric enumeration problems.

Findings

Constructed spectral curves for four geometric problems

Unified approach links unstable geometries to spectral data

Facilitates computation of enumerative invariants

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.