Counting lattice points in compactified moduli spaces of curves

Do, Norman; Norbury, Paul

doi:10.2140/gt.2011.15.2321

Cited by 14 publications

(47 citation statements)

References 13 publications

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“…His proof is similar in that it involved an edge removal operation, but the main formula and its proof therein contained are incorrectly recorded -the terms involving products of functions N g,n were double counted and need a compensating factor of 1 2 . A corrected version appears in [10,41]. Our proof presented here is new, and is based on a different idea using ciliation.…”

Section: Definition 32mentioning

confidence: 99%

The Kontsevich constants for the volume of the moduli of curves and topological recursion

Chapman

Mulase

Safnuk

2011

Communications in Number Theory and Physics

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We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon graphs. As an application we obtain a new proof of the Kontsevich constants for the ratio of the Euclidean and the symplectic volumes of the moduli space of curves.MSC Primary: 14N35, 05C30, 53D30, 11P21; Secondary: 81T30 1. The Witten-Kontsevich theory for the tautological cotangent class (i.e. the ψ-class) 2 The combinatorial model of the moduli space Let us begin with reviewing basic facts about ribbon graphs and the combinatorial model of the moduli space M g,n due to Harer [22], Mumford [38], and Strebel [47]. We refer to [35] for precise definitions and more detailed exposition.A ribbon graph of topological type (g, n) is the 1-skeleton of a cell-decomposition of a closed oriented topological surface Σ of genus g that decomposes the surface into a disjoint union of v 0-cells, e 1-cells, and n 2-cells. The Euler characteristic of the surface is given by 2 − 2g = v − e + n. The 1-skeleton of a cell-decomposition is a graph Γ drawn on Σ,

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Section: Definition 32mentioning

confidence: 99%

The Kontsevich constants for the volume of the moduli of curves and topological recursion

Chapman

Mulase

Safnuk

2011

Communications in Number Theory and Physics

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“…However, they are not sufficient. For instance, N 7 3,2 (2n + 1, 1) = N 3,2,n+2 (2n + 1, 1) = 0, but max(0, 6 − n) = max(k, 2g + n − 1 − 1 2 b i ) ≤ 7 = t ≤ min(3g − 3 + n, 2g + n − 1) = min (8,7). To see why, suppose there were such an arc diagram; then there must be a (necessarily non-separating) arc connecting the two boundary components.…”

Section: Inequalities On Regionsmentioning

confidence: 99%

Counting curves on surfaces

Koyama

Mathews

2017

Int. J. Math.

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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.

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“…For the spectral curve x(z) = z + 1 z and y(z) = z, the dilaton equation was used to define the enumeration of ribbon graphs where some of the b i are equal to zero [26]. This was a crucial ingredient in the enumeration of lattice points in the Deligne-Mumford compactification of moduli spaces of curves [9].…”

Section: Bmmentioning

confidence: 99%

Quantum curves for the enumeration of ribbon graphs and hypermaps

Manescu

2014

Communications in Number Theory and Physics

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Abstract. The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-commutative quantisation whose associated differential operator annihilates the partition function for the spectral curve. In this paper, we determine the quantum curves and partition functions for an infinite sequence of enumerative problems involving generalisations of ribbon graphs known as hypermaps. These results give rise to an explicit conjecture relating hypermap enumeration to the topological recursion and we provide evidence to support this conjecture.

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Counting lattice points in compactified moduli spaces of curves

Cited by 14 publications

References 13 publications

The Kontsevich constants for the volume of the moduli of curves and topological recursion

The Kontsevich constants for the volume of the moduli of curves and topological recursion

Counting curves on surfaces

Quantum curves for the enumeration of ribbon graphs and hypermaps

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