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The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures

Motohico Mulase, Sergey Shadrin, Loek Spitz

Abstract

We derive the spectral curves for $q$-part double Hurwitz numbers, $r$-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0,1)-geometry. We quantize this family of spectral curves and obtain the Schroedinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.

The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures

Abstract

We derive the spectral curves for -part double Hurwitz numbers, -spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0,1)-geometry. We quantize this family of spectral curves and obtain the Schroedinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.

Paper Structure

This paper contains 16 sections, 4 theorems, 58 equations, 2 tables.

Table of Contents

  1. Introduction and the main results
  2. Hurwitz numbers and Eynard-Orantin recursion
  3. Generalizations of Hurwitz numbers
  4. Spectral curves
  5. Schrödinger equations
  6. Organization of the paper
  7. Infinite-wedge space
  8. $q$-Double Hurwitz numbers
  9. Spectral curve from $(0,1)$ geometry
  10. Principal specialization
  11. Quantization
  12. $r$-Spin Hurwitz numbers
  13. Spectral curve from $(0,1)$ geometry
  14. Principal specialization
  15. Quantization
  16. ...and 1 more sections

Key Result

Proposition \oldthetheorem

Denote by $\zeta$ the function Then we have in particular, and, taking a limit as $z \to 0$, Note that the proposition is still true when we replace any of the $\mathcal{E}$-operators on the left-hand side by the corresponding $\tilde{\mathcal{E}}$.

Theorems & Definitions (13)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • ...and 3 more