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The Laplace Transform and Quantum Curves

Quinten Weller

Abstract

A Laplace transform that maps the topological recursion (TR) wavefunction to its $x$-$y$ swap dual is defined. This transform is then applied to the construction of quantum curves. General results are obtained, including a formula for the quantisation of many spectral curves of the form $e^xP_2(e^y) - P_1(e^y) = 0$ where $P_1$ and $P_2$ are coprime polynomials; an important class that contains interesting spectral curves related to mirror symmetry and knot theory that have, heretofore, evaded the general TR-based methods previously used to derive quantum curves. Quantum curves known in the literature are reproduced, and new quantum curves are derived. In particular, the quantum curve for the $T$-equivariant Gromov-Witten theory of $\mathbb{P}(a,b)$ is obtained.

The Laplace Transform and Quantum Curves

Abstract

A Laplace transform that maps the topological recursion (TR) wavefunction to its - swap dual is defined. This transform is then applied to the construction of quantum curves. General results are obtained, including a formula for the quantisation of many spectral curves of the form where and are coprime polynomials; an important class that contains interesting spectral curves related to mirror symmetry and knot theory that have, heretofore, evaded the general TR-based methods previously used to derive quantum curves. Quantum curves known in the literature are reproduced, and new quantum curves are derived. In particular, the quantum curve for the -equivariant Gromov-Witten theory of is obtained.

Paper Structure

This paper contains 20 sections, 12 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Outline
  5. Acknowledgments
  6. Background
  7. Spectral curves and topological recursion
  8. Quantising spectral curves
  9. $x$-$y$ swap duality
  10. The Laplace transform
  11. Quantum curves
  12. General consequences of the Laplace transform
  13. Examples
  14. $P(x,y) = e^x + e^{(f+1)y} - e^{fy} = 0$
  15. $P(x,y) = c^P e^x(e^{y}-c^2)^Q - c^Q e^{Py}(e^{y}-1)^Q = 0$
  16. ...and 5 more sections

Key Result

Theorem 2.7

The differentials $\omega_{g,n}$ enjoy the following properties:

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Definition 2.8
  • Definition 2.9
  • ...and 34 more