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Quantum Curves in the Context of Symplectic Duality

Alexander Hock, Sergey Shadrin

TL;DR

The paper develops a unified framework for deriving quantum curves from topological recursion by leveraging universal x–y duality and symplectic duality. Starting from the spectral data $(\Sigma,x,y,B)$, it shows how various TR generalizations (CN-TR, BE-TR, Log-TR, Gen-TR) yield wave functions and associated quantum curves, with genus-zero cases admitting finite $\hbar$-corrections. It provides explicit constructions and a broad catalog of examples (Airy, Bessel, Hurwitz/logarithmic cases, and torus knots) which illustrate how dualities simplify or restructure the quantization process and illuminate base-point dependence. The work demonstrates that Gen-TR, in particular, can produce quantum curves beyond the standard TR families, via concrete transformations and kernel-duality relations, thereby offering a versatile tool for quantization in enumerative geometry, knot theory, and related areas.

Abstract

We discuss how to use the recent progress in understanding of the $x$-$y$ duality and symplectic duality in the theory of topological recursion and its generalizations in order to efficiently compute the quantum spectral curve operators for the wave functions with arbitrary base points. The paper also contains an overview of recent generalizations of the setup of topological recursion prompted by the progress in understanding the $x$-$y$ duality.

Quantum Curves in the Context of Symplectic Duality

TL;DR

The paper develops a unified framework for deriving quantum curves from topological recursion by leveraging universal x–y duality and symplectic duality. Starting from the spectral data , it shows how various TR generalizations (CN-TR, BE-TR, Log-TR, Gen-TR) yield wave functions and associated quantum curves, with genus-zero cases admitting finite -corrections. It provides explicit constructions and a broad catalog of examples (Airy, Bessel, Hurwitz/logarithmic cases, and torus knots) which illustrate how dualities simplify or restructure the quantization process and illuminate base-point dependence. The work demonstrates that Gen-TR, in particular, can produce quantum curves beyond the standard TR families, via concrete transformations and kernel-duality relations, thereby offering a versatile tool for quantization in enumerative geometry, knot theory, and related areas.

Abstract

We discuss how to use the recent progress in understanding of the - duality and symplectic duality in the theory of topological recursion and its generalizations in order to efficiently compute the quantum spectral curve operators for the wave functions with arbitrary base points. The paper also contains an overview of recent generalizations of the setup of topological recursion prompted by the progress in understanding the - duality.

Paper Structure

This paper contains 39 sections, 9 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgments
  4. Background
  5. CEO-TR
  6. Extensions of topological recursion
  7. Irregular topological recursion
  8. Higher order topological recursion
  9. Logarithmic topological recursion
  10. Generalized topological recursion
  11. Quantum curve from topological recursion
  12. Symplectic transformations
  13. Higher genus
  14. Non-algebraic curves and Log-TR
  15. Towards Gen-TR
  16. ...and 24 more sections

Key Result

Theorem 2.17

Let $\{\omega_{g,n}\}$ be generated by CEO-TR. If the underlying spectral curve has genus $0$, then there exists a quantum curve $\hat{P}_\hbar (\hat{x}, \hat{y})$ that annihilates the wave function $\psi_{x(z_0)}(x(z))$, i.e., where the quantum curve $\hat{P}_\hbar (\hat{x}, \hat{y})$ has only finitely many $\hbar$-corrections and has in the semiclassical limit $\lim_{\hbar \to 0} \hat{P}_\hbar

Figures (1)

  • Figure 1: A schematic overview of the different versions of topological recursion. The most recent extension, Gen-TR, reproduces CEO-TR, Log-TR and CN-TR in general and BE-TR under some additional assumption.

Theorems & Definitions (44)

  • Definition 2.1: Eynard:2007kz
  • Remark 2.2
  • Remark 2.3
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • Remark 2.10
  • Remark 2.11
  • ...and 34 more