Moduli Spaces Of Quadratic Differentials Research Articles

Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab–Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichmüller disk of a quadratic differential lying in this connected component.

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.

Tau Function and Moduli of Meromorphic Quadratic Differential

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

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.

Stability conditions and braid group actions on affine An quivers

We study stability conditions on the Calabi–Yau-[Formula: see text] categories associated to an affine type [Formula: see text] quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order [Formula: see text]. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type [Formula: see text] based on the Khovanov–Seidel–Thomas method.

RECURSION FOR MASUR-VEECH VOLUMES OF MODULI SPACES OF QUADRATIC DIFFERENTIALS

AbstractWe derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

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.

Hodge and Prym Tau Functions, Strebel Differentials and Combinatorial Model of $${\mathcal {M}}_{g,n}$$

The goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Strebel differentials. These line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $$\mathcal {L}_j$$ , and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $$\kappa _1$$ -circle bundle. By evaluating the increment of the phase around co-dimension 2 sub-complexes, we identify the Poincare dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten’s cycle $$W_{5} $$ and Kontsevich’s boundary. This provides combinatorial analogues of Mumford’s relations on $${\mathcal {M}}_{g,n}$$ and Penner’s relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $$W_{5} $$ are interpreted as pentagon moves while those of loops around Kontsevich’s boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Strebel differentials, parametrized in terms of period, or homological coordinates; we show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter as the intersection pairing in the odd homology of the canonical double cover.

Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.

Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Chen-Moeller-Zagier on the long-standing conjecture about the large genus asymptotics of Masur-Veech volumes, we derive that the relative contribution is asymptotically of the order 1/d, where d is the dimension of the stratum. Similarly, we evaluate the contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes of low-dimensional strata in the moduli space of quadratic differentials. We combine this count with our recent result on equidistribution of one-cylinder square-tiled surfaces translated to the language of interval exchange transformations to compute empirically approximate values of the Masur-Veech volumes of strata of quadratic differentials of all small dimensions.

Principal boundary of moduli spaces of abelian and quadratic differentials

The seminal work of Eskin–Masur–Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne–Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur–Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge–Chen–Gendron–Grushevsky–Möller.

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 moduli spaces of quadratic differentials: getting explicit values

The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quadratic differentials with at most simple poles in all low dimensions.

Right-angled billiards and volumes of moduli spaces of quadratic differentials on CP^1

We use the relation between the volumes of the strata of mero- morphic quadratic differentials with at most simple poles on CP 1 and counting functions of the number of (bands of) closed geodesics in associated flat met- rics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. Applying ergodic techniques to the Teichmuller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.

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.

The pillowcase distribution and near-involutions

In the context of the Eskin-Okounkov approach to the calculation of the volumes of the different strata of the moduli space of quadratic differentials, the important ingredients are the pillowcase weight probability distribution on the space of Young diagrams, and the asymptotic study of characters of permutations that near-involutions. In this paper we present various new results for these objects. Our results give light to unforeseen difficulties in the general solution to the problem, and they simplify some of the previous proofs.

Winning games for bounded geodesics in moduli spaces of quadratic differentials

We prove that the set of bounded geodesics in Teichmuller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmuller disk and for intervals exchanges with any fixed irreducible permutation.

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.

Ends of strata of the moduli space of quadratic differentials

Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short saddle connections, but the discrete data associated to this splitting can be quite difficult to describe. In order to bypass these difficulties, we use the Veech zippered rectangles construction and the corresponding (extended) Rauzy classes.

Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials

In this paper, we study fundamental groups of strata of the moduli space of quadratic differentials. We use certain properties of the Abel-Jacobi map, combined with local surgeries on quadratic differentials, to construct quotient groups of the fundamental groups for a particular family of strata.