Counting curves on surfaces

TL;DR

This paper explores an elementary combinatorial problem of counting disjoint arcs on surfaces with boundary points, revealing a rich structure connected to topological recursion, algebraic geometry, and mathematical physics.

Contribution

It introduces a new combinatorial enumeration problem for curves on surfaces, demonstrating recursive formulas, quasi-polynomial behavior, and links to advanced geometric and physical theories.

Findings

Curve counts follow an effective recursion via topological recursion.

Enumerative counts exhibit quasi-polynomial behavior.

Generating functions show structures akin to free energies and partition functions.

Abstract

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour. This "elementary curve-counting" is in fact related to a more advanced notion of "curve-counting" from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Furthermore, among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions, topological recursion, and quantum curves.