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

Understand

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.