Meromorphic Differentials Research Articles

Connected components of strata of residueless meromorphic differentials

Connected components of strata of residueless meromorphic differentials

Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case

In this paper, we study the spectral theory of soliton condensates - a special limit of soliton gases - for the focusing NLS (fNLS). In particular, we analyze the kinetic equation for the fNLS circular condensate, which represents the first example of an explicitly solvable fNLS condensate with nontrivial large scale space-time dynamics. Solution of the kinetic equation was obtained by reducing it to Whitham type equations for the endpoints of spectral arcs. We also study the rarefaction and dispersive shock waves for circular condensates, as well as calculate the corresponding average conserved quantities and the kurtosis. We want to note that one of the main objects of the spectral theory - the nonlinear dispersion relations - is introduced in the paper as some special large genus (thermodynamic) limit the Riemann bilinear identities that involve the quasimomentum and the quasienergy meromorphic differentials.

The Schwarzian Derivative and the Degree of a Classical Minimal Surface

Using the Schwarzian derivative we construct a sequence ( P l ) l ⩾ 2 of meromorphic differentials on every non-flat oriented minimal surface in Euclidean 3-space. The differentials ( P l ) l ⩾ 2 are invariant under all deformations of the surface arising via the Weierstrass representation and depend on the induced metric and its derivatives only. A minimal surface is said to have degree n if its n-th differential is a polynomial expression in the differentials of lower degree. We observe that several well-known minimal surfaces have small degree, including Enneper’s surface, the helicoid/catenoid and the Scherk—as well as the Schwarz family. Furthermore, it is shown that locally and away from umbilic points every minimal surface can be approximated by a sequence of minimal surfaces of increasing degree.

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.

Period realization of meromorphic differentials with prescribed invariants

Period realization of meromorphic differentials with prescribed invariants

Airy structures and deformations of curves in surfaces

AbstractAn embedded curve in a symplectic surface defines a smooth deformation space of nearby embedded curves. A key idea of Kontsevich and Soibelman is to equip the symplectic surface with a foliation in order to study the deformation space . The foliation, together with a vector space of meromorphic differentials on , endows an embedded curve with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on . Kontsevich and Soibelman define an Airy structure on to be a formal quadratic Lagrangian which leads to an alternative construction of the tensors of topological recursion. In this paper, we produce a formal series on which takes it values in , and use this to produce the Donagi–Markman cubic from a natural cubic tensor on , giving a generalisation of a result of Baraglia and Huang.

Confocal Families of Hyperbolic Conics via Quadratic Differentials

We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics.

Complete Solution of the LSZ Model via Topological Recursion

We prove that the Langmann-Szabo-Zarembo (LSZ) model with quartic potential, a toy model for a quantum field theory on noncommutative spaces grasped as a complex matrix model, obeys topological recursion of Chekhov, Eynard and Orantin. By introducing two families of correlation functions, one corresponding to the meromorphic differentials $\omega_{g,n}$ of topological recursion, we obtain Dyson-Schwinger equations that eventually lead to the abstract loop equations being, together with their pole structure, the necessary condition for topological recursion. This strategy to show the exact solvability of the LSZ model establishes another approach towards the exceptional property of integrability in some quantum field theories. We compare differences in the loop equations for the LSZ model (with complex fields) and the Grosse-Wulkenhaar model (with hermitian fieldss) and their consequences for the resulting particular type of topological recursion that governs the models.

Equations of linear subvarieties of strata of differentials

For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-M\"oller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere.

The boundary of linear subvarieties

We describe the boundary of linear subvarieties in the moduli space of multi-scale differentials. Linear subvarieties are algebraic subvarieties of strata of (possibly) meromorphic differentials that in local period coordinates are given by linear equations. The main example of such are affine invariant submanifolds, that is, closures of \operatorname{SL}(2,\mathbb{R}) -orbits. We prove that the boundary of any linear subvariety is again given by linear equations in generalized period coordinates of the boundary. Our main technical tool is an asymptotic analysis of periods near the boundary of the moduli space of multi-scale differentials which yields further techniques and results of independent interest.

Recent developments in spectral theory of the focusing NLS soliton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases

In this paper we consider soliton and breather gases for one dimensional integrable focusing nonlinear Schrödinger equation (fNLS). We derive average densities and fluxes for such gases by studying the thermodynamic limit of the fNLS finite gap solutions. Thermodynamic limits of quasimomentum, quasienergy and their connections with the corresponding g-functions were also established. We then introduce the notion of periodic fNLS gases and calculate for them the average densities, fluxes and thermodynamic limits of meromorphic differentials. Certain accuracy estimates of the obtained results are also included. Our results constitute another step towards the mathematical foundation for the spectral theory of fNLS soliton and breather gases that appeared in work of El and Tovbis (2020 Phys. Rev. E 101 052207).

On the locus of genus 3 curves that admit meromorphic differentials with a zero of order 6 and a pole of order 2

The main goal of this article is to compute the class of the divisor of ℳ ¯ 3 obtained by taking the closure of the image of Ωℳ 3 (6;-2) by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to ℳ g and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge–Chen–Gendron–Grushevsky–Möller.

Translation surfaces and periods of meromorphic differentials

Let S $S$ be an oriented surface of genus g $g$ and n $n$ punctures. The periods of any meromorphic differential on S $S$ , with respect to a choice of complex structure, determine a representation χ : Γ g , n → C $\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where Γ g , n $\Gamma _{g,n}$ is the first homology group of S $S$ . We characterise the representations that thus arise, that is, lie in the image of the period map Per : Ω M g , n → Hom ( Γ g , n , C ) $\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$ . This generalises a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on S $S$ with the prescribed holonomy χ $\chi$ . Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.

𝒩 = 1 Curves on Generalized Coulomb Branches of Supersymmetric Gauge Theories

We study the low energy effective dynamics of four-dimensional N=1 superconformal theories on their generalized Coulomb branch. The low energy effective gauge couplings are naturally encoded in algebraic curves X, which we derive for general values of the couplings and mass deformations. We then recast these IR curves X to the UV or M-theory form C: the punctured Riemann surfaces on which the M5 branes are compactified giving the four-dimensional theories. We find that the UV curves C and their corresponding meromorphic differentials take the same form as those for their mother four-dimensional N=2 theories of class S. They have the same poles, and their residues are functions of all the exactly marginal couplings and the bare mass parameters which we can compute exactly.

Towards a classification of connected components of the strata of $k$-differentials

A k -differential on a Riemann surface is a section of the k -th power of the canonical bundle. Loci of k -differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of k -differentials. The classification of connected components of the strata of k -differentials was known for holomorphic differentials, meromorphic differentials and quadratic differentials with at worst simple poles by Kontsevich-Zorich, Boissy and Lanneau, respectively. Built on their work we develop new techniques to study connected components of the strata of k -differentials for general k . As an application, we give a complete classification of connected components of the strata of quadratic differentials with arbitrary poles. Moreover, we distinguish certain components of the strata of k -differentials by generalizing the hyperelliptic structure and spin parity for higher k . We also describe an approach to determine explicitly parities of k -differentials in genus zero and one, which inspires an amusing conjecture in number theory. A key viewpoint we use is the notion of multi-scale k -differentials introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller for k = 1 and extended by Costantini-Möller-Zachhuber for all k .

The closure of double ramification loci via strata of exact differentials

Double ramification loci, also known as strata of 0 -differentials, are algebraic subvarieties of the moduli space of smooth curves parametrizing Riemann surfaces such that there exists a rational function with prescribed ramification over 0 and \infty . We describe the closure of double ramification loci inside the Deligne–Mumford compactification in geometric terms. To a rational function we associate its exact differential, which allows us to realize double ramification loci as linear subvarieties of strata of meromorphic differentials. We then obtain a geometric description of the closure using our recent results on the boundary of linear subvarieties. Our approach yields a new way of relating the geometry of loci of rational functions and Teichmüller dynamics. We also compare our results to a different approach using admissible covers.

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.

On existence of quasi-Strebel structures for meromorphic $k$-differentials

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k\ge 4 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k -differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine–Stieltjes theory of linear differential equations with polynomial coefficients.

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.