In Search of a Hidden Curve

TL;DR

This paper explores how identifying algebraic curves, or spectral curves, can solve enumeration problems in geometry and combinatorics, revealing polynomial structures and connections to quantum geometric quantization.

Contribution

It demonstrates the role of spectral curves in solving enumeration problems, providing new proofs and insights, and discusses the open problem of finding the spectral curve for Apéry sequences.

Findings

Polynomiality of Hurwitz number generating functions.

Short proofs of Witten-Kontsevich and Faber-Pandharipande theorems.

Connection between spectral curves and quantization in geometric contexts.

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 $\lambda_g$-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 Ap\'ery 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.