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Quantum curves from refined topological recursion: the genus 0 case

Omar Kidwai, Kento Osuga

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.

Quantum curves from refined topological recursion: the genus 0 case

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.
Paper Structure (24 sections, 25 theorems, 153 equations, 1 table)

This paper contains 24 sections, 25 theorems, 153 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Topological recursion
  4. Refined topological recursion and $\beta$-deformation
  5. Quantum curves
  6. Main results
  7. Comments and open problems
  8. Refined topological recursion
  9. Unrefined topological recursion
  10. Refined topological recursion
  11. Refined spectral curves and the refined recursion
  12. Reduction to the unrefined case
  13. Properties of refined topological recursion
  14. Proof of RTR\ref{['RTR1']}
  15. Proof of RTR\ref{['RTR2']}, RTR\ref{['RTR3']}, and RTR\ref{['RTR4']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\mathcal{S}^{\bm \mu}$ be a genus zero degree two refined spectral curve satisfying Assumption ass:tr. For any $2g+n\geq2$, the multidifferentials constructed from the refined topological recursion satisfy the following properties: where $J=(p_1,\ldots,p_n)\in \Sigma^n$, $\mathcal{R}^*$ is a certain subset of $\mathcal{R}$, and $\Phi$ is any primitive of $ydx$.

Theorems & Definitions (57)

  • Theorem 1.1: Theorem \ref{['thm:RTR']}
  • Theorem 1.2: Theorem \ref{['thm:main']}
  • Definition 2.1
  • Definition 2.3: CEOEOEO2
  • Remark 2.4
  • Theorem 2.5: EO
  • Definition 2.6
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 47 more