Topological recursion of Eynard-Orantin and the Harmonic Oscillator

TL;DR

This paper applies the Eynard-Orantin topological recursion to the harmonic oscillator curve, linking it to the WKB wave function and generating Poincaré polynomials for orbifolds of metric ribbon graphs.

Contribution

It demonstrates the equivalence of topological recursion results with the WKB wave function for the harmonic oscillator and connects to orbifold Poincaré polynomials.

Findings

Topological recursion reproduces the WKB wave function.

Multi-differentials generate Poincaré polynomials for orbifolds.

Establishes a link between recursion, quantum mechanics, and geometric invariants.

Abstract

We apply the Chekhov-Eynard-Orantin topological recursion to the curve corresponding to the quantum harmonic oscillator and demonstrate that the result is equivalent to the WKB wave function. We also show that using the multi-differentials obtained by the topological recursion from the harmonic oscillator curve, one generates naturally the so-called Poincar\'e polynomials associated with the orbifolds of the metric ribbon graphs.