The Eynard-Orantin recursion and equivariant mirror symmetry for the projective line

TL;DR

This paper demonstrates that the Eynard-Orantin recursion applied to an equivariant mirror Landau-Ginzburg model of the projective line encodes all genus, all descendant equivariant Gromov-Witten invariants, linking several key conjectures in enumerative geometry.

Contribution

It establishes a comprehensive recursion framework for equivariant Gromov-Witten invariants of the projective line, connecting mirror symmetry and enumerative geometry conjectures.

Findings

Eynard-Orantin recursion encodes all genus, all descendants equivariant Gromov-Witten invariants.

Non-equivariant limit confirms the Norbury-Scott conjecture.

Large radius limit recovers the Bouchard-Marino conjecture on simple Hurwitz numbers.

Abstract

We study the equivariantly perturbed mirror Landau-Ginzburg model of the projective line. We show that the Eynard-Orantin recursion on this model encodes all genus all descendants equivariant Gromov-Witten invariants of the projective line. The non-equivariant limit of this result is the Norbury-Scott conjecture, while by taking large radius limit we recover the Bouchard-Marino conjecture on simple Hurwitz numbers.