Meromorphic Quadratic Differentials Research Articles

Delay Painlevé-I equation, associated polynomials and Masur-Veech volumes

We study a delay-differential analogue of the first Painlevé equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlevé-I equation in terms of an affine Weyl group of type A1(1). As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas.

Schwarzian Versus a Family of Moving Parabolic Points

In this article, the trajectories of meromorphic quadratic differentials whose coefficients are Schwarzian derivatives of rational maps are studied. We conjecture that the graphs consisting of the so-called negative-real critical Schwarzian trajectories are finite for all rational maps of degree at least two. For any given rational map f, we construct a family of parabolic rational maps { g w } w having a parabolic fixed point at the non-critical point w of f. If w is a zero of the Schwarzian Sf , we prove that the orientations of the negative-real critical trajectories of S f ( z ) d z 2 landing at w coincide with the attracting directions of gw at w. The critical curve in every immediate parabolic basin of w is an arc starting at w along the attracting direction and ending at a critical point of f, which is a pole of Sf . We conduct some numerical experiments to compare the critical curves in the immediate parabolic basins with the negative-real critical Schwarzian trajectories, which support our conjecture.

Quadratic differentials of real algebraic curves

It was shown by J. C. Langer and D. A. Singer in their influential 2007 paper [5] that real algebraic curves can be interpreted as trajectories of meromorphic quadratic differentials defined on appropriate Riemann surfaces. The goal of this note is to suggest a more elementary approach to this problem that is based on the classical complex analysis. To demonstrate how our approach works, we apply it to the simplest non-trivial case of real algebraic curves; i.e. to conics.

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.

WKB expansion for a Yang–Yang generating function and the Bergman tau function

We study symplectic properties of the monodromy map of second-order equations on a Riemann surface whose potential is meromorphic with double poles. We show that the Poisson bracket defined in terms of periods of the meromorphic quadratic differential implies the Goldman Poisson structure on the monodromy manifold. We apply these results to a WKB analysis of this equation and show that the leading term in the WKB expansion of the generating function of the monodromy symplectomorphism (the Yang–Yang function introduced by Nekrasov, Rosly, and Shatashvili) is determined by the Bergman tau function on the moduli space of meromorphic quadratic differentials.

Quadratic differentials, measured foliations, and metric graphs on punctured surfaces

A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole singularities. In a neighborhood of a pole, such a foliation comprises foliated strips and half-planes, and its leaf space determines a metric graph. We introduce the notion of an asymptotic direction at each pole and show that for a punctured surface equipped with a choice of such asymptotic data, any compatible pair of measured foliations uniquely determines a complex structure and a meromorphic quadratic differential realizing that pair. This proves the analogue of a theorem of Gardiner–Masur for meromorphic quadratic differentials. We also prove an analogue of the Hubbard–Masur theorem; namely, for a fixed punctured Riemann surface there exists a meromorphic quadratic differential with any prescribed horizontal foliation, and such a differential is unique provided we prescribe the singular flat geometry at the poles.

Wild globally hyperbolic maximal anti‐de Sitter structures

Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their deformation space: as a quotient of the product of two copies of the Teichm\"uller space of crowned hyperbolic surfaces and as the bundle over the Teichm\"uller space of $\Sigma$ of meromorphic quadratic differentials with poles of order at least $3$ at the punctures.

Veech groups of flat surfaces with poles

Flat surfaces that correspond to meromorphic 1-forms with poles or to meromorphic quadratic differentials containing poles of order two and higher have infinite flat area. We classify groups that appear as Veech groups of translation surfaces with poles. We characterize those surfaces such that their GL + (2,ℝ)-orbit or their SL(2,ℝ)-orbit is closed. Finally, we provide a way to determine the Veech group for a typical infinite surface in any given chamber of a stratum.

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.

Eigenvalues of the Thurston operator

Let f : C ̂ → C ̂ be a postcritically finite rational map, and let Q ( C ̂ ) be the space of meromorphic quadratic differentials on C ̂ with simple poles. We study the set of eigenvalues of the pushforward operator f ∗ : Q ( C ̂ ) → Q ( C ̂ ) . In particular, we show that when f : C → C is a unicritical polynomial of degree D with periodic critical point, the eigenvalues of f ∗ : Q ( C ̂ ) → Q ( C ̂ ) are contained in the annulus 1 4 D < | λ | < 1 and belong to 1 D U where U is the group of algebraic units.

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.

Regular globally hyperbolic maximal anti‐de Sitter structures

Let Σ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on Σ × R and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of Σ and as the bundle over the Teichmüller space of Σ whose fibre consists of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.

Harmonic maps and wild Teichmüller spaces

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

Discriminant Circle Bundles over Local Models of Strebel Graphs and Boutroux Curves

We study special circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$. The space $\mathcal Q_0^{\mathbb R}(-7)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order 7 with real periods; it appears naturally in the study of a neighbourhood of the Witten's cycle $W_1$ in the combinatorial model based on Jenkins-Strebel quadratic differentials of $\mathcal M_{g,n}$. The space $\mathcal Q^{\mathbb R}_0([-3]^2)$ is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most 3 with real periods; it appears in description of a neighbourhood of Kontsevich's boundary $W_{-1,-1}$ of the combinatorial model. The application of the formalism of the Bergman tau-function to the combinatorial model (with the goal of computing analytically Poincare dual cycles to certain combinations of tautological classes) requires the study of special sections of circle bundles over $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$; in the case of the space $\mathcal Q_0^{\mathbb R}(-7)$ a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces $\mathcal Q_0^{\mathbb R}(-7)$ and $\mathcal Q^{\mathbb R}_0([-3]^2)$, also called the spaces of Boutroux curves, in detail, together with corresponding circle bundles.

Meromorphic quadratic differentials and measured foliations on a Riemann surface

We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential q if we prescribe, in addition, the principal parts of \(\sqrt{q}\) at the poles. This generalizes a theorem of Hubbard and Masur for holomorphic quadratic differentials. The proof analyzes infinite-energy harmonic maps from the Riemann surface to \(\mathbb {R}\)-trees of infinite co-diameter, with prescribed behavior at the poles.

The Cantor–Bendixson Rank of Certain Bridgeland–Smith Stability Conditions

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor–Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor–Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a “slit translation surface” is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.

Quadratic differentials, half-plane structures, and harmonic maps to trees

Let (\Sigma,p) be a pointed Riemann surface and k\geq 1 an integer. We parametrize the space of meromorphic quadratic differentials on \Sigma with a pole of order k+2 at p , having a connected critical graph and an induced metric composed of k Euclidean half-planes. The parameters form a finite-dimensional space \mathcal{L} \cong \mathbb{R}^{k} \times S^1 that describe a model singular-flat metric around the puncture with respect to a choice of coordinate chart. This generalizes an important theorem of Strebel, and associates, to each point in \mathcal{T}_{g,1} \times \mathcal{L} , a unique metric spine of the surface that is a ribbon-graph with k infinite-length edges to p . The proofs study and relate the singular-flat geometry of the quadratic differential, and the infinite-energy harmonic map from \Sigma \setminus p to a k -pronged tree, having the same Hopf differential.

Stability conditions on $$\text {CY}_N$$ CY N categories associated to $$A_n$$ A n -quivers and period maps

In this paper, we study the space of stability conditions on a certain N-Calabi-Yau ( $$\mathrm {CY}_N$$ ) category associated to an $$A_n$$ -quiver. Recently, Bridgeland and Smith constructed stability conditions on some $$\mathrm {CY}_3$$ categories from meromorphic quadratic differentials with simple zeros. Generalizing their results to higher dimensional Calabi-Yau categories, we describe the space of stability conditions as the universal cover of the space of polynomials of degree $$n+1$$ with simple zeros. In particular, central charges of stability conditions on $$\mathrm {CY}_N$$ categories are constructed as the periods of quadratic differentials with zeros of order $$N-2$$ which are associated to polynomials.

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.