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In Search of a Hidden Curve

Motohico Mulase

TL;DR

The notes argue that many geometric-counting problems are governed by hidden spectral curves, enabling a unifying workflow: identify a spectral curve from the simplest invariants, apply a Laplace transform to obtain polynomial generating data, and then perform a semiclassical quantization to obtain quantum curves and opers. This framework yields concrete results: Catalan numbers produce a spectral curve $x = z + \frac{1}{z}$ with Laurent-polynomial $F_{g,n}$ and connections to $\chi(\mathcal{M}_{g,n})$ and the Witten–Kontsevich theory, while Hurwitz numbers correspond to the Lambert curve $x = y e^{-y}$ and admit a topological recursion that recovers the DVV constraints and $\lambda_g$-formulas; Apéry’s $\zeta(3)$ story remains open for a spectral-curve interpretation. The exposition ties these successes to Higgs bundles, opers, and Deligne’s $\hbar$-connections, illustrating how a spectral curve serves as a universal invariant from which higher-genus data can be generated and studied via mirror-symmetric (Laplace-transform) methods. Overall, the work highlights a deep correspondence among combinatorics, moduli spaces, and integrable systems, with the Laplace transform acting as a concrete realization of a mirror operation in this geometric setting.

Abstract

It has been noticed since around 2007 that certain enumeration problems can be solved when an analytic or algebraic curve is identified. This curve is the key to the problem. In these lectures, a few such examples are presented. One is a detailed account on counting simple Hurwitz numbers, explaining how the problem was solved by discovering this key curve. The formula for the curve allows us to write the generating functions of Hurwitz numbers in terms of polynomials. This unexpected polynomiality produces, as a byproduct, straightforward and short proofs of the Witten-Kontsevich theorem and the $λ_g$-theorem of Faber-Pandharipande. An analogous enumeration problem associated with Catalan numbers is also presented, which has a simpler feature in terms of analysis. The asymptotic behavior of counting leads this time to the Euler characteristic of the moduli spaces of smooth curves. We then discuss another enumeration problem, the Apéry sequences. The quest of identifying the hidden curve for this case remains open. These curves, also known as spectral curves, are discovered via solving ordinary differential equations. The counting problem of geometric origin associated with the genus 0, one marked point case is encoded in the spectral curve. It is explained that going from the $(g,n)=(0,1)$-case to arbitrary $(g,n)$ is a process of quantization of the spectral curve. This perspective of quantization is discussed in a geometric setting, when the differential equations are linear with holomorphic coefficients, in terms of Higgs bundles, opers, and Gaiotto's conformal limit construction. In this context, however, there are no counting problems behind the scene.

In Search of a Hidden Curve

TL;DR

The notes argue that many geometric-counting problems are governed by hidden spectral curves, enabling a unifying workflow: identify a spectral curve from the simplest invariants, apply a Laplace transform to obtain polynomial generating data, and then perform a semiclassical quantization to obtain quantum curves and opers. This framework yields concrete results: Catalan numbers produce a spectral curve with Laurent-polynomial and connections to and the Witten–Kontsevich theory, while Hurwitz numbers correspond to the Lambert curve and admit a topological recursion that recovers the DVV constraints and -formulas; Apéry’s story remains open for a spectral-curve interpretation. The exposition ties these successes to Higgs bundles, opers, and Deligne’s -connections, illustrating how a spectral curve serves as a universal invariant from which higher-genus data can be generated and studied via mirror-symmetric (Laplace-transform) methods. Overall, the work highlights a deep correspondence among combinatorics, moduli spaces, and integrable systems, with the Laplace transform acting as a concrete realization of a mirror operation in this geometric setting.

Abstract

It has been noticed since around 2007 that certain enumeration problems can be solved when an analytic or algebraic curve is identified. This curve is the key to the problem. In these lectures, a few such examples are presented. One is a detailed account on counting simple Hurwitz numbers, explaining how the problem was solved by discovering this key curve. The formula for the curve allows us to write the generating functions of Hurwitz numbers in terms of polynomials. This unexpected polynomiality produces, as a byproduct, straightforward and short proofs of the Witten-Kontsevich theorem and the -theorem of Faber-Pandharipande. An analogous enumeration problem associated with Catalan numbers is also presented, which has a simpler feature in terms of analysis. The asymptotic behavior of counting leads this time to the Euler characteristic of the moduli spaces of smooth curves. We then discuss another enumeration problem, the Apéry sequences. The quest of identifying the hidden curve for this case remains open. These curves, also known as spectral curves, are discovered via solving ordinary differential equations. The counting problem of geometric origin associated with the genus 0, one marked point case is encoded in the spectral curve. It is explained that going from the -case to arbitrary is a process of quantization of the spectral curve. This perspective of quantization is discussed in a geometric setting, when the differential equations are linear with holomorphic coefficients, in terms of Higgs bundles, opers, and Gaiotto's conformal limit construction. In this context, however, there are no counting problems behind the scene.
Paper Structure (27 sections, 19 theorems, 214 equations, 6 figures)

This paper contains 27 sections, 19 theorems, 214 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The story begins with Catalan numbers and their Laplace transform
  3. A miraculous integer sequence
  4. The unique quantization for the holomorphic cases
  5. A quick guide of the contents
  6. Prelude
  7. Theme $1$: Mirror symmetry of Catalan numbers
  8. Theme $2$: When is a space a moduli space?
  9. Theme $3$: Finding a curve from a period matrix
  10. Theme $4$: The $\tau$-functions
  11. The spectral curve for Hurwitz numbers
  12. The Laplace transform
  13. Simple Hurwitz numbers
  14. The Laplace transform of the simple Hurwitz numbers
  15. Two quantum curves for the Lambert curve
  16. ...and 12 more sections

Key Result

Theorem 1.4

For the case of $2g=2+n>0$, substitute each $x_i$ in Fgn Catalan with $$ and write the result as $F_{g,n}(t_1,\dots,t_n)$ incorporating these substitutions. Then for every $(g,n)$ in the range of $2g-2+n>0$, the following holds:

Figures (6)

  • Figure 1: The mirror dual spectral curve $\Sigma$ of the Catalan numbers.
  • Figure 2: A cell-decomposition of a curve of genus $0$ with one $0$-cell. The arrow on a half-edge indicates no rotation symmetry is allowed.
  • Figure 1.1: $1$-Skeleton of a Cell Decomposition and a Cell Graph
  • Figure 2.1: A new point $x_{\ell+1}$ is attached on top of $x_3$ (left), and at nodal point $x$ (right).
  • Figure 2.2: Construction of a based tree from two rooted trees.
  • ...and 1 more figures

Theorems & Definitions (50)

  • Remark 2
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: CMSOM4OM5DMSSMP2012
  • Theorem 1.5: M1984
  • Theorem 2.2: BMEMSMZ
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 40 more