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Quantization of classical spectral curves via topological recursion

Bertrand Eynard, Elba Garcia-Failde, Olivier Marchal, Nicolas Orantin

Abstract

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

Quantization of classical spectral curves via topological recursion

Abstract

We prove that the topological recursion formalism can be used to quantize any generic classical spectral curve with smooth ramification points and simply ramified away from poles. For this purpose, we build both the associated quantum curve, i.e.~the differential operator quantizing the algebraic equation defining the classical spectral curve considered, and a basis of wave functions, that is to say a basis of solutions of the corresponding differential equation. We further build a Lax pair representing the resulting quantum curve and thus present it as a point in an associated space of meromorphic connections on the Riemann sphere, a first step towards isomonodromic deformations. We finally propose two examples: the derivation of a 2-parameter family of formal trans-series solutions to Painlevé 2 equation and the quantization of a degree three spectral curve with pole only at infinity.

Paper Structure

This paper contains 92 sections, 31 theorems, 554 equations.

Table of Contents

  1. Introduction
  2. Topological recursion and quantum curves
  3. Main results
  4. Organization of the article
  5. General notations
  6. Admissible spectral curves
  7. Classical spectral curves
  8. Local coordinates, spectral times and admissible initial data
  9. Canonical local coordinates
  10. Spectral times
  11. Bergman kernel
  12. Topological recursion
  13. Definition
  14. Properties of differentials produced by the topological recursion
  15. Ramification points at poles
  16. ...and 77 more sections

Key Result

Lemma 3.1

Let $(\omega_{h,n}')_{h,n\geq 0}$ be the topological recursion differential forms defined by taking residues at all $a\in\mathcal{R}_0$ (i.e. all ramification points, so including $a\in x^{-1}(\mathcal{P})$). If, for all ramification points $p\in x^{-1}(\mathcal{P})$, we have $r_p\geq 3$ and $t_{p,r

Theorems & Definitions (112)

  • Definition 2.1: Classical spectral curves
  • Definition 2.2: Classical spectral curves with fixed pole structure
  • Definition 2.3: Curve punctured at the poles
  • Definition 2.4: Ramification points and critical values
  • Definition 2.5: Admissible classical spectral curves
  • Definition 2.6: Canonical local coordinates
  • Definition 2.7: Spectral times
  • Definition 2.8: Local potentials
  • Definition 2.9: Bergman kernel
  • Definition 2.10: Admissible initial data
  • ...and 102 more