Uniformizing Map Research Articles

Thermal corrections to Rényi entropy in BMS field theory

In the study of three-dimensional flat holography, the BMS field theory manifests the infinite-dimensional BMS3 symmetry, a powerful tool in elucidating numerous universal phenomena. This paper explores a certain low-temperature limit of the BMS field theory. The primary focus lies in the calculation of the thermal correction to the Rényi entropy of the single interval on the cylinder from the replica trick and the uniformizing map. As a double check, an alternative method calculating the entanglement entropy is introduced, with the entanglement first law and the modular Hamiltonian.

Uniformizing Lee-Yang singularities

Motivated by the search for the QCD critical point, we discuss how to obtain the singular behavior of a thermodynamic system near a critical point, namely the Lee-Yang singularities, from a limited amount of local data generated in a different region of the phase diagram. We show that by using a limited number of Taylor series coefficients, it is possible to reconstruct the equation of state past the radius of convergence, in particular in the critical region. Furthermore we also show that it is possible to extend this reconstruction to go from a crossover region to the first-order transition region in the phase diagram, using a uniformizing map to pass between Riemann sheets. We illustrate these ideas via the Chiral Random Matrix Model and the Ising Model.

Berry phases in the reconstructed KdV equation.

We consider the KdV equation on a circle and its Lie-Poisson reconstruction, which is reminiscent of an equation of motion for fluid particles. For periodic waves, the stroboscopic reconstructed motion is governed by an iterated map whose Poincaré rotation number yields the drift velocity. We show that this number has a geometric origin: it is the sum of a dynamical phase, a Berry phase, and an "anomalous phase." The last two quantities are universal: they are solely due to the underlying Virasoro group structure. The Berry phase, in particular, was previously described by Oblak [J. High Energy Phys. 10, 114 (2017)] for two-dimensional conformal field theories and follows from adiabatic deformations produced by the propagating wave. We illustrate these general results with cnoidal waves, for which all phases can be evaluated in closed form thanks to a uniformizing map that we derive. Along the way, we encounter "orbital bifurcations" occurring when a wave becomes non-uniformizable: there exists a resonance wedge, in the cnoidal parameter space, where particle motion is locked to the wave, while no such locking occurs outside of the wedge.

Equivalent descriptions of the loewner energy

Loewner’s equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil–Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil–Petersson metric on the Weil–Petersson Teichmüller space.

Density of algebraic points on Noetherian varieties

Let $${\Omega \subset \mathbb{R}^{n}}$$ be a relatively compact domain. A finite collection of real valued functions on $${{\Omega}}$$ is called a Noetherian chain if the partial derivatives of each function are expressible as polynomials in the functions. A Noetherian function is a polynomial combination of elements of a Noetherian chain. We introduce Noetherian parameters (degrees, size of the coefficients) which measure the complexity of a Noetherian chain. Our main result is an explicit form of the Pila–Wilkie theorem for sets defined using Noetherian equalities and inequalities: for any $${\varepsilon > 0}$$ , the number of points of height H in the transcendental part of the set is at most C· $${{H}^ \varepsilon}$$ where C can be explicitly estimated from the Noetherian parameters and $${\varepsilon}$$ . We show that many functions of interest in arithmetic geometry fall within the Noetherian class, including elliptic and abelian functions, modular functions and universal covers of compact Riemann surfaces, Jacobi theta functions, periods of algebraic integrals, and the uniformizing map of the Siegel modular variety $${\mathcal{A}_{g}}$$ . We thus effectivize the (geometric side of) Pila–Zannier strategy for unlikely intersections in various contexts.

An o-minimal characterization of special subvarieties

Let S = Γ ∖ X + be a connected component of a Shimura variety and let π : X + ⟶ S be the uniformizing map. We show that an algebraic subvariety V of S is weakly special if and only if an irreducible analytic component of π - 1 V is definable in an o-minimal structure over R.

Applications du théorème d’Ax–Lindemann hyperbolique

AbstractWe explain how the André–Oort conjecture for a general Shimura variety can be deduced from the hyperbolic Ax–Lindemann conjecture, a good lower bound for Galois orbits of special points and the definability, in the$o$-minimal structure${ \mathbb{R} }_{\mathrm{an} , \mathrm{exp} } $, of the restriction to a fundamental set of the uniformizing map of a Shimura variety. These ingredients are known in some important cases. As a consequence a proof of the André–Oort conjecture for projective special subvarieties of${ \mathcal{A} }_{6}^{N} $for an arbitrary integer$N$is given.

Riemann surfaces out of paper

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S.

Contractibility of the space of rational maps

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space \(\operatorname {Bun}_{G}\) of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.

Constructing conformal maps of triangulated surfaces

We describe a means of computing the uniformizing conformal map from a triangulated surface whose triangles are realized as Euclidean triangles in R3 onto a fundamental domain in the unit disc D, plane C, or sphere S2. Mapping such triangulated surfaces arises in a number of applications, such as conformal brain flattening. We use the circle packing technique of Bowers, Hurdal, Stephenson, et al., to first create a quasiconformal approximation to the conformal map; then we apply a discrete form of conformal welding to reduce the distortion and converge to the conformal map in the limit.

Almost sure multifractal spectrum for the tip of an SLE curve

The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

Paper surfaces and dynamical limits

It is very common in mathematics to construct surfaces by identifying the sides of a polygon together in pairs: For example, identifying opposite sides of a square yields a torus. In this article the construction is considered in the case where infinitely many pairs of segments around the boundary of the polygon are identified. The topological, metric, and complex structures of the resulting surfaces are discussed: In particular, a condition is given under which the surface has a global complex structure (i.e., is a Riemann surface). In this case, a modulus of continuity for a uniformizing map is given. The motivation for considering this construction comes from dynamical systems theory: If the modulus of continuity is uniform across a family of such constructions, each with an iteration defined on it, then it is possible to take limits in the family and hence to complete it. Such an application is briefly discussed.

Liouville theory and uniformization of four-punctured sphere

A few years ago Zamolodchikov and Zamolodchikov proposed an expression for the four-point classical Liouville action in terms of the three-point actions and the classical conformal block [Nucl. Phys. B 477, 577 (1996)]. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.

Computing the Laurent Series of the Map Ψ: C − [formula omitted] → C − M

We discuss several interesting properties of the Laurent series of Ψ : C − D → C − M, the inverse of the uniformizing map of the Mandelbrot set M = { c ∈ C : c, c 2 + c,( c 2 + c) 2 + c, . , ↛ ∞ as n → ∞}. Continuity of the Laurent series on ∂ D implies local connectivity of M, which is an open question. We show how the coefficients of the series can he easily computed by following Hubbard and Douady′s construction of the uniformizing map for M. As a result, we show that the coefficients are rational with powers of 2 in their denominator and that many are zero. Furthermore, if the series is continuous on ∂D, we show that it is not Hölder continuous. We also include several empirical observations made by Don Zagier on the growth of the power of 2 in the denominator.