Counting curves on surfaces
Abstract
In this paper we consider an elementary, and largely unexplored, combinatorial problem in lowdimensional topology. Consider a real 2dimensional 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 quasipolynomial behaviour. This "elementary curvecounting" is in fact related to a more advanced notion of "curvecounting" 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 quasipolynomials 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.08853
 Bibcode:
 2015arXiv151208853D
 Keywords:

 Mathematics  Geometric Topology;
 Mathematical Physics;
 Mathematics  Combinatorics;
 05A15;
 81S10;
 57M50
 EPrint:
 87 pages, 11 figures. v2: Included references to previous work in the literature