Quantum curves from refined topological recursion: the genus 0 case

TL;DR

This paper develops a refined topological recursion for genus zero degree two curves, establishing fundamental properties, connecting to quantum curves, and simplifying forms in the Nekrasov-Shatashvili limit.

Contribution

It introduces a new geometric formulation of refined topological recursion for genus zero curves, extending existing theories and providing explicit quantum curve formulas.

Findings

Proved fundamental properties of the refined recursion for genus zero curves.

Generalized the quantization of spectral curves to this refined setting.

Derived simplified quantum curves in the Nekrasov-Shatashvili limit.

Abstract

We formulate geometrically (without reference to physical models) a refined topological recursion applicable to genus zero curves of degree two, inspired by Chekhov-Eynard and Marchal, introducing new degrees of freedom in the process. For such curves, we prove the fundamental properties of the recursion analogous to the unrefined case. We show the quantization of spectral curves due to Iwaki-Koike-Takei can be generalized to this setting and give the explicit formula, which turns out to be related to the unrefined case by a simple transformation. For an important collection of examples, we write down the quantum curves and find that in the Nekrasov-Shatashvili limit, they take an especially simple form.