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From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

Bertrand Eynard, Elba Garcia-Failde

Abstract

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.

From topological recursion to wave functions and PDEs quantizing hyperelliptic curves

Abstract

Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree spectral curves with a global involution satisfy a system of partial differential equations, whose equations can be seen as quantizations of the original spectral curves. The families of spectral curves can be parametrized with the so-called times, defined as periods on second type cycles, and with the poles. These equations can be used to prove that the WKB solution of many isomonodromic systems coincides with the topological recursion wave function, which proves that the topological recursion wave function is annihilated by a quantum curve. This recovers many known quantum curves for genus zero spectral curves and generalizes this construction to hyperelliptic curves.

Paper Structure

This paper contains 27 sections, 14 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Quantum curves and topological recursion
  3. Generalized cycles
  4. Context and outline
  5. Comparison to the literature
  6. Outline
  7. Spectral curves with a global involution
  8. Spectral curves as input of the topological recursion
  9. Spectral curves from isomonodromic systems
  10. Motivational example: The first Painlevé equation
  11. Parametrizations and deformation parameters of the elliptic case
  12. Degenerate case
  13. Non-degenerate case: elliptic curves
  14. Loop equations and deformation parameters
  15. Variational formulas of the topological recursion
  16. ...and 12 more sections

Key Result

Theorem 3.1

BEO Let $g, n\in \mathbb{N}$. The linear loop equations read: The quadratic loop equations claim that the following expression is a rational function of $x(z)$ with no poles at the branch-points.

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • Corollary 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 27 more