Counting curves on surfaces

Abstract

In this paper we consider an elementary, and largely unexplored, combinatorial problem in lowdimensional 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.

Counting Non-Crossing Permutations on Surfaces of any Genus

“…In this paper we show that the count of polygon diagrams has an analogous structure. The arguments here parallel those in [9], which deal with the enumeration of arc diagrams. Our arguments are necessarily more involved than those for arc diagrams but, perhaps surprisingly, produce cleaner results.…”

“…. , µ n ) behave similarly not only to arc diagram counts [9], but also to the volume polynomials of Kontsevich [21], the Weil-Petersson volume polynomials of Mirzakhani [23], and the lattice count polynomials of Norbury [25]. These last three problems are all governed by the topological recursion of Chekhov, Eynard and Orantin [4,12] and the fact that they agree to leading order is a consequence of this fact.…”

“…As the name suggests, the relationship between P g,n and Q g,n mirrors that of dessin and pruned dessin counts [11], as well as Hurwitz numbers and pruned Hurwitz numbers [10]. It also mirrors the relationship between the counts of arc diagrams G g,n and non-boundaryparallel arc diagrams N g,n in [9]; we can think of the latter as pruned arc diagrams. This correspondence between combinatorial objects and their pruned counterparts also bears a resemblance to constructions appearing in the literature on maps, such as the notion of non-choppable maps [2].…”

“…We also need some results for summing polynomials over tuples of integers satisfying constraints on their sum and parities. They can be proved as in [9] using generalisations of Ehrhart's theorem from [3], but we give more elementary proofs in the appendix.…”

“…If all polygons in D are bigons, then collapsing them into arcs turns D into an arc diagram, as defined and studied by the first and third authors with Koyama in [9]. The count of arc diagrams exhibits quasi-polynomial behaviour, and the asymptotic behaviour is governed by intersection numbers on moduli spaces of curves.…”