The Eynard--Orantin recursion for simple singularities

TL;DR

This paper extends the local Eynard--Orantin recursion to a global form for simple singularities, linking spectral curves to Weyl group invariants and relating free energies to singularity potentials.

Contribution

It proves the extension of local to global recursion for simple singularities and connects spectral curves with Weyl group invariants, also relating free energies to singularity potentials.

Findings

Global recursion extends local Eynard--Orantin recursion for simple singularities.

Spectral curve is defined by Weyl group invariant polynomials.

Genus 0 and 1 free energies match singularity potentials up to constants.

Abstract

According to \cite{BOSS} and \cite{M1}, the ancestor correlators of any semi-simple cohomological field theory satisfy {\em local} Eynard--Orantin recursion. In this paper, we prove that for simple singularities, the local recursion can be extended to a global one. The spectral curve of the global recursion is an interesting family of Riemann surfaces defined by the invariant polynomials of the corresponding Weyl group. We also prove that for genus 0 and 1, the free energies introduced in \cite{EO} coincide up to some constant factors with respectively the genus 0 and 1 primary potentials of the simple singularity.