Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

Abstract

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space Qg,n of genus g meromorphic quadratic differentials with at most n simple poles and no other poles as polynomials in the intersection numberswith explicit rational coefficients, where g ′ < g and n ′ < 2g + n. The formulae obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomialsalso appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces M g ′ ,n ′ (b 1 , . . . , b n ′ ) of bordered hyperbolic surfaces with geodesic boundaries of lengths b 1 , . . . , b n ′ .A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by M. Mirzakhani through a completely different approach. We prove a further result: the density of the mapping class group orbit Modg,n •γ of any simple closed multicurve γ inside the ambient set MLg,n(Z) of integral measured laminations computed by Mirzakhani, coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to γ among all square-tiled surfaces in Qg,n.We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case when n = 0. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are 2 3πg • 1 4 g times less frequent.

Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow

“…The integral points of the PIL structure on SH + (λ) correspond to integer multicurves transverse to λ, so when λ is itself a multicurve, integral points correspond to square-tiled surfaces. Using our coordinates for F uu (λ), one can recover the leading coefficient for the polynomial counting the number of square-tiled surfaces with given horizontal curve of bounded area (which was originally computed in [AH20a,DGZZ20]). In particular, since renormalized lattice point counts equidistribute to Lebesgue measure, the coefficient in question can be identified as the Lebesgue measure of (a torus bundle over) the portion of the combinatorial moduli space B(S \ λ)/ Mod(S \ λ) with controlled boundary lengths.…”

Reconstructing GKZ via Topological Recursion

“…In this approach, the combinatorial data of the distribution of simple closed geodesics is described by stable graphs, and the Masur-Veech volumes are computed by combinations of Weil-Petersson volumes. In recent years, Mirzakhani's combinatorial approach to compute the Masur-Veech volumes is established further in a series of works by Delecroix, Goujard, Zograf, and Zorich [21,22,23,24,25]. (See also [44,17] for the related works.)…”

“…In particular for the principal stratum of the moduli space Q g,n of quadratic differentials, a novel connection between the Delecroix-Goujard-Zograf-Zorich's result [25] and the ABO topological recursion is proposed in [10,8]. The ABO topological recursion to compute the Masur-Veech volumes is the Laplace dual of the CEO topological recursion for the Airy spectral curve accompanied with an action of twist.…”

“…And it is natural to consider the analogue of the Masur-Veech volume for these models in Table 2. In this article, we will compute the analogue of the Masur-Veech volume for each 2D gravity model by the combinatorial method developed in [69,25] and the twisted CEO topological recursion (1.15). This paper is organized as follows.…”

An Explicit Formula for Witten’s 2-Correlators

“…Remark. A careful analysis of the numbers b g,k performed in [1], Sect. 4, implies that for any g ≥ 1 and k = 2, .…”