Siegel Veech Constants Research Articles

Masur–Veech volumes and intersection theory: The principal strata of quadratic differentials

We describe a conjectural formula via intersection numbers for the Masur–Veech volumes of strata of quadratic differentials with prescribed zero orders, and we prove the formula for the case when the zero orders are odd. For the principal strata of quadratic differentials with simple zeros, the formula reduces to computing the top Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. An appendix proves that the intersection of this class with ψ-classes can be computed by Eynard–Orantin topological recursion. As applications, we analyze numerical properties of Masur–Veech volumes, area Siegel–Veech constants, and sums of Lyapunov exponents of the principal strata for fixed genus and varying number of zeros, which settles the corresponding conjectures due to Grivaux and Hubert, Fougeron, and elaborated in Andersen et al. We also describe conjectural formulas for area Siegel–Veech constants and sums of Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them for the principal strata.

Topological recursion for Masur–Veech volumes

We study the Masur–Veech volumes M V g , n $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g $g$ with n $n$ punctures. We show that the volumes M V g , n $MV_{g,n}$ are the constant terms of a family of polynomials in n $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [Delecroix, Goujard, Zograf, Zorich, Duke J. Math 170 (2021), no. 12, math.GT/1908.08611] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [Andersen, Borot, Orantin, Geometric recursion, math.GT/1711.04729, 2017]. We also obtain an expression of the area Siegel–Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur–Veech volumes, and thus of area Siegel–Veech constants, for low g $g$ and n $n$ , which leads us to propose conjectural formulae for low g $g$ but all n $n$ . We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials

In this paper we analyze the large genus asymptotics for intersection numbers between $\psi$-classes, also called correlators, on the moduli space of stable curves. Our proofs proceed through a combinatorial analysis of the recursive relations (Virasoro constraints) that uniquely determine these correlators, together with a comparison between the coefficients in these relations with the jump probabilities of a certain asymmetric simple random walk. As an application of this result, we provide the large genus limits for Masur-Veech volumes and area Siegel-Veech constants associated with principal strata in the moduli space of quadratic differentials. These confirm predictions of Delecroix-Goujard-Zograf-Zorich from 2019.

Pillowcase covers: counting Feynman-like graphs associated with quadratic differentials

We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity results of Eskin-Okounkov and a practical method to compute area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality result for 2-orbifold Hurwitz numbers with completed cycles.

Large genus asymptotics for volumes of strata of abelian differentials

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume ν 1 ( H 1 ( m ) ) \nu _1 \big ( \mathcal {H}_1 (m) \big ) of a stratum indexed by a partition m = ( m 1 , m 2 , … , m n ) m = (m_1, m_2, \ldots , m_n) is ( 4 + o ( 1 ) ) ∏ i = 1 n ( m i + 1 ) − 1 \big ( 4 + o(1) \big ) \prod _{i = 1}^n (m_i + 1)^{-1} , as 2 g − 2 = ∑ i = 1 n m i 2g - 2 = \sum _{i = 1}^n m_i tends to ∞ \infty . This confirms a prediction of Eskin-Zorich and generalizes some of the recent results of Chen-Möller-Zagier and Sauvaget, who established these limiting statements in the special cases m = 1 2 g − 2 m = 1^{2g - 2} and m = ( 2 g − 2 ) m = (2g - 2) , respectively. We also include an appendix by Anton Zorich that uses our main result to deduce the large genus asymptotics for Siegel-Veech constants that count certain types of saddle connections.

Masur–Veech volumes of quadratic differentials and their asymptotics

Based on the Chen–Möller–Sauvaget formula, we apply the theory of integrable systems to derive three equations for the generating series of the Masur–Veech volumes VolQg,n associated with the principal strata of the moduli spaces of quadratic differentials, and propose refinements of the conjectural formulas given in Delecroix et al. [11] and Aggarwal et al. [4] for the large genus asymptotics of VolQg,n and of the associated area Siegel–Veech constants.

Trimmed sums of twists and the area Siegel\u2013Veech constant

We relate trimmed sums of twists in cylinders along a typical Teichmüller geodesic to the area Siegel–Veech constant.

Conjectural Large Genus Asymptotics of Masur–Veech Volumes and of Area Siegel–Veech Constants of Strata of Quadratic Differentials

We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity.

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

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 numbers ∫M‾ g′,n′ψ1d1⋯ψ n′dn′ with explicit rational coefficients, where g′<g and n′<2g+n. The formulas obtained in this article are derived from lattice point counts involving the Kontsevich volume polynomials N g′,n′(b1,…,bn′) that also appear in Mirzakhani’s recursion for the Weil–Petersson volumes of the moduli spaces Mg′,n′(b1,…,bn′) of bordered hyperbolic surfaces with geodesic boundaries of lengths b1,…,bn′. A similar formula for the Masur–Veech volume (but without explicit evaluation) was obtained earlier by 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 nonseparating 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 4g times less frequent.

Large Genus Asymptotics for Siegel–Veech Constants

In this paper we consider the large genus asymptotics for two classes of Siegel–Veech constants associated with an arbitrary connected stratum $$\mathcal {H} (\alpha )$$ of Abelian differentials. The first is the saddle connection Siegel–Veech constant $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big )$$ counting saddle connections between two distinct, fixed zeros of prescribed orders $$m_i$$ and $$m_j$$ , and the second is the area Siegel–Veech constant $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big )$$ counting maximal cylinders weighted by area. By combining a combinatorial analysis of explicit formulas of Eskin–Masur–Zorich that express these constants in terms of Masur–Veech strata volumes, with a recent result for the large genus asymptotics of these volumes, we show that $$c_{\mathrm{sc}}^{m_i, m_j} \big ( \mathcal {H} (\alpha ) \big ) = (m_i + 1) (m_j + 1) \big ( 1 + o(1) \big )$$ and $$c_{\mathrm{area}} \big ( \mathcal {H}(\alpha ) \big ) = \frac{1}{2} + o(1)$$ , both as $$|\alpha | = 2g - 2$$ tends to $$\infty $$ . The former result confirms a prediction of Zorich and the latter confirms one of Eskin–Zorich in the case of connected strata.

Volumes and Siegel–Veech constants of $${\mathcal{H}}$$H (2G − 2) and Hodge integrals

In the 80’s H. Masur and W. Veech defined two numerical invariants of strata of abelian differentials: the volume and the Siegel–Veech constant. Based on numerical experiments, A. Eskin and A. Zorich proposed a series of conjectures for the large genus asymptotics of these invariants. By a careful analysis of the asymptotic behavior of quasi-modular forms, D. Chen, M. Moeller, and D. Zagier proved that this conjecture holds for strata of differentials with simple zeros. Here, with a mild assumption of existence of a good metric, we show that the conjecture holds for the other extreme case, i.e. for strata of differentials with a unique zero. Our main ingredient is the expression of the numerical invariants of these strata in terms of Hodge integrals on moduli spaces of curves.

Quasimodularity and large genus limits of Siegel-Veech constants

Quasimodular forms were first studied systematically in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants. In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We also introduce certain modified Siegel-Veech weights whose generating functions will later be shown to be quasimodular. Parts II and III are devoted to the study of the (quasi) modular properties of the generating functions arising from weighted counting of torus coverings. These two parts contain little geometry and can be read independently of the rest of the paper. The starting point is the theorem of Bloch and Okounkov saying that certain weighted averages, called q q -brackets, of shifted symmetric functions on partitions are quasimodular forms. In Part II we give an expression for the growth polynomials (a certain polynomial invariant of quasimodular forms) of these q q -brackets in terms of Gaussian integrals, and we use this to obtain a closed formula for the generating series of cumulants that is the basis for studying large genus asymptotics. In Part III we show that the even hook-length moments of partitions are shifted symmetric polynomials, and we prove a surprising formula for the q q -bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial as a linear combination of derivatives of Eisenstein series. This formula gives a quasimodularity statement also for the ( − 2 ) (-2) -nd hook-length moments by an appropriate extrapolation, and this in turn implies the quasimodularity of the Siegel-Veech weighted counting functions. Finally, in Part IV these results are used to give explicit generating functions for the volumes and Siegel-Veech constants in the case of the principal stratum of abelian differentials. The generating functions have an amusing form in terms of the inversion of a power series (with multiples of Bernoulli numbers as coefficients) that gives the asymptotic expansion of a Hurwitz zeta function. To apply these exact formulas to the Eskin-Zorich conjectures on large genus asymptotics (both for the volume and the Siegel-Veech constant) we provide in a separate appendix a general framework for computing the asymptotics of rapidly divergent power series.

Counting problem on wind-tree models

We study periodic wind-tree models, billiards in the plane endowed with $\mathbb{Z}^2$-periodically located identical connected symmetric right-angled obstacles. We show asymptotic formulas for the number of (isotopy classes of) closed billiard trajectories (up to $\mathbb{Z}^2$-translations) on the wind-tree billiard. We also compute explicitly the associated Siegel-Veech constant for generic wind-tree billiards depending on the number of corners on the obstacle.

Convergence of Siegel–Veech constants

We show that for any weakly convergent sequence of ergodic \(SL_2(\mathbb {R})\)-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel–Veech constants converge to the Siegel–Veech constant of the limit measure. Together with a measure equidistribution result due to Eskin–Mirzakhani–Mohammadi, this yields the (previously conjectured) convergence of sequences of Siegel–Veech constants associated to Teichmuller curves in genus two. The proof uses a recurrence result closely related to techniques developed by Eskin–Masur. We also use this recurrence result to get an asymptotic quadratic upper bound, with a uniform constant depending only on the stratum, for the number of saddle connections of length at most R on a unit-area translation surface.

Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials

For a non-uniform lattice in SL (2,\mathbb R) , we consider excursions of a random geodesic in cusp neighborhoods of the quotient finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. In the Teichmüller setting, we consider invariant measures for the SL (2,\mathbb R) action on the moduli spaces of quadratic differentials. By the work of Eskin and Mirzakhani, these measures are supported on affine invariant submanifolds of a stratum of quadratic differentials. For a Teichmüller geodesic random with respect to a SL (2,\mathbb R) -invariant measure, we study its excursions in thin parts of the associated affine invariant submanifold. Under a regularity hypothesis for the invariant measure, we prove similar strong laws for certain partial sums involving these excursions. The limits in these laws are related to the volume asymptotic of the thin parts. By Siegel–Veech theory, these are given by various Siegel–Veech constants. As a direct consequence, we show that the word metric of mapping classes that approximate a Teichmüller geodesic ray that is random with respect to the Masur–Veech measure, grows faster than T log T .

Volumes of Strata of Abelian Differentials and Siegel–Veech Constants in Large Genera

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.

Siegel–Veech Constants for Strata of Moduli Spaces of Quadratic Differentials

We present an explicit formula relating volumes of strata of meromorphic quadratic differentials with at most simple poles on Riemann surfaces and counting functions of the number of flat cylinders filled by closed geodesics in associated flat metric with singularities. This generalizes the result of Athreya, Eskin and Zorich in genus 0 to higher genera.

Geometry of periodic regions on flat surfaces and associated Siegel–Veech constants

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel–Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel–Veech constants associated to the geometry of configurations of periodic cylinders and their area, and study extremal properties of such configurations in a fixed stratum and in all strata of a fixed genus.

Covers of the projective line and the moduli space of quadratic differentials

Consider the Hurwitz space parameterizing covers of \({\mathbb{P}^1}\) branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmuller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmuller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.

Degenerations of quadratic differentials on ℂℙ1

We describe the connected components of the complement of a natural "diagonal" of real codimension 1 in a stratum of quadratic differentials on CP1. We establish a natural bijection between the set of these connected components and the set of generic configurations that appear on such "flat spheres". We also prove that the stratum has only one topological end. Finally, we elaborate a necessary toolkit destined to evaluation of the Siegel-Veech constants.