Quantum curves and topological recursion

TL;DR

This survey explores the connection between quantum curves, represented by Schrödinger-like operators, and topological recursion, highlighting how they encode quantum enumerative invariants and can be constructed via different methods.

Contribution

It provides a comprehensive overview of how quantum curves relate to topological recursion and discusses their role in encoding quantum enumerative invariants.

Findings

Quantum curves are noncommutative analogues of plane curves.

Topological recursion can be used to construct wave functions associated with quantum curves.

The Schrödinger operator annihilates the wave function constructed via WKB or topological recursion.

Abstract

This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schr\"odinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schr\"odinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion.