Two-dimensional Torus Research Articles

Light fermion masses in partially deconstructed models

Considering a theory space consisting of a large number of five-dimensional Dirac fermion field theories including background abelian gauge fields, we can construct a theory similar to a continuous six-dimensional theory compactified with two-dimensional manifolds with and without magnetic flux or orbifolds as extra dimensions. This method, called dimensional deconstruction, can be used to construct a model with one-dimensional discrete space, which represents general graph structures. In this paper, we propose the models with two extra dimensions, which resemble two-dimensional tori, cylinders, and rectangular regions, as continuum limits. We also try to build a model that mimics one with the two-dimensional orbifold compactification.

Deformational rigidity of integrable metrics on the torus

Abstract It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.

On the boundedness of solutions of a forced discontinuous oscillator

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

On non-trivial hyperbolic sets and their bifurcations in families of diffeomorphisms of a two-dimensional torus.

We propose a simple model-two-parameter family of diffeomorphisms of a two-dimensional torus. Combining analytical and numerical methods, we find regions in the parameter plane such that each diffeomorphism of the family is hyperbolic and describe the structure of the corresponding hyperbolic sets. We also study bifurcations on the boundaries of these regions, which lead to the change of hyperbolicity type (from Anosov diffeomorphisms to DA-diffeomorphisms).

Study of self-oscillations of a distributed rotor from a material with hereditarily elastic properties

Self-oscillatory solutions of dynamic equations of transverse vibrations of a distributed rotating ideally balanced rotor (shaft) on an isotropic support made of a material with nonlinear hereditary properties are studied. The derivation of dynamic equations (mathematical model), which are a system of two nonlinear partial differential equations with an infinite (integral) delay of the time argument, is given. A mathematical formulation of the initial–boundary value problem is formulated, its solvability and correctness are proven. The stability of the zero solution (stable rotation), the mechanisms of loss of stability, and the self-oscillatory solutions bifurcating in this case are studied. The possibility of bifurcation of periodic solutions (direct asynchronous precessions) and invariant two-dimensional tori (beat modes) is shown. The stability of self-oscillating solutions is determined by the parameters of the problem under consideration. The method of central manifolds of distributed systems and the theory of bifurcations were used as a research method. Asymptotic formulas are constructed for self-oscillating solutions.

Isosystolic inequalities on two-dimensional Finsler tori

In this article we survey all known optimal isosystolic inequalities on two-dimensional Finsler tori involving the following two central notions of Finsler area: the Busemann–Hausdorff area and the Holmes–Thompson area. We also complete the panorama by establishing the following new optimal isosystolic inequality that is deduced from prior work by Burago and Ivanov: the Busemann–Hausdorff area of a Finsler reversible 2 -torus with unit systole is at least equal to \pi/4 .

The ALE partition functions of M-strings

We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings’ worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.

L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus

AbstractWe consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.

Exact Controllability and Stabilization for Linear Dispersive PDE’s on the Two-Dimensional Torus

Exact Controllability and Stabilization for Linear Dispersive PDE’s on the Two-Dimensional Torus

Strong convergence of the vorticity and conservation of the energy for the α-Euler equations

In this paper, we study the convergence of solutions of the α-Euler equations to solutions of the Euler equations on the two-dimensional torus. In particular, given an initial vorticity ω 0 in Lxp for p∈(1,∞) , we prove strong convergence in Lt∞Lxp of the vorticities q α , solutions of the α-Euler equations, towards a Lagrangian and energy-conserving solution of the Euler equations. Furthermore, if we consider solutions with bounded initial vorticity, we prove a quantitative rate of convergence of q α to ω in Lp , for p∈(1,∞) .

Strichartz estimates and global well-posedness of the cubic NLS on

Abstract The optimal $L^4$ -Strichartz estimate for the Schrödinger equation on the two-dimensional rational torus $\mathbb {T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in $H^s(\mathbb {T}^2)$ for any $s>0$ and data that are small in the critical norm.

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

MHD equilibria with nonconstant pressure in nondegenerate toroidal domains

We prove the existence of piecewise smooth MHD equilibria in three-dimensional toroidal domains of \mathbb R^3 where the pressure is constant on the boundary but not in the interior. The pressure is piecewise constant and the plasma current exhibits an arbitrary number of current sheets. We also establish the existence of free boundary steady states surrounded by vacuum with an external surface current. The toroidal domains where these equilibria are shown to exist need not be small perturbations of an axisymmetric domain, and in fact they can have any knotted topology. The building blocks we use in our construction are analytic toroidal domains satisfying a certain nondegeneracy condition, which roughly states that there exists a force-free field that is ergodic on the surface of the domain. The proof involves three main ingredients: a gluing construction of piecewise smooth MHD equilibria, a Hamilton–Jacobi equation on the two-dimensional torus that can be understood as a nonlinear deformation of the cohomological equation (so the nondegeneracy assumption plays a major role in the corresponding analysis), and a new KAM theorem tailored for the study of divergence-free fields in three dimensions whose Poincaré map cannot be computed explicitly.

Discrete Rössler Oscillators: Maps and Their Ensembles

We study the complex dynamics of a discrete analogue of the classical flow dynamical system — Rössler oscillator. Minimal ensembles of two and three coupled discrete oscillators with different topologies are considered. As the main research tool we used the method of Lyapunov exponents charts. For coupled systems, the possibility of two-, three- and four-frequency quasi-periodicity is revealed. Illustrations in the form of Fourier spectra are presented. Doublings of invariant curves, two- and three-dimensional tori are found. The transition from two-dimensional tori to three-dimensional ones occurs through a quasi-periodic saddle-node bifurcation of invariant tori or through a quasi-periodic Hopf bifurcation. A discrete version of the hyperchaotic Rössler oscillator is also discussed. It exhibits dynamical behavior close to a flow system in some measure.

Invariant Gibbs dynamics for the two-dimensional Zakharov-Yukawa system

We study the Gibbs dynamics for the Zakharov-Yukawa system on the two-dimensional torus T2, namely a Schrödinger-wave system with a Zakharov-type coupling (−Δ)γ. We first construct the Gibbs measure in the weakly nonlinear coupling case (0≤γ<1). Combined with the non-construction of the Gibbs measure in the strongly nonlinear coupling case (γ=1) by Oh, Tolomeo, and the author (2020), this exhibits a phase transition at γ=1. We also study the dynamical problem and prove almost sure global well-posedness of the Zakharov-Yukawa system and invariance of the Gibbs measure under the resulting dynamics for the range 0≤γ<13. In this dynamical part, the main step is to prove local well-posedness. Our argument is based on the first order expansion and the operator norm approach via the random matrix/tensor estimate from a recent work Deng, Nahmod, and Yue (2020). In the appendix, we briefly discuss the Hilbert-Schmidt norm approach and compare it with the operator norm approach.

Multiscale Coupling and the Maximum of {mathcal {P}}(phi )_2 Models on the Torus

We establish a coupling between the {mathcal {P}}(phi )_2 measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied phi ^4_2 measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué–Dupuis stochastic control representation for {mathcal {P}}(phi )_2. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field.As an application of the coupling, we prove that the maximum of the {mathcal {P}}(phi )_2 field on the discretised torus with mesh-size epsilon > 0 converges in distribution to a randomly shifted Gumbel distribution as epsilon rightarrow 0.

Almost representations of algebras and quantization

abstract: We introduce the notion of almost representations of Lie algebras and quantum tori, and establish an Ulam-stability type phenomenon: every irreducible almost representation is close to a genuine irreducible representation. As an application, we prove that geometric quantizations of the two-dimensional sphere and the two-dimensional torus are conjugate in the semi-classical limit up to a small error.

On Some Properties of Semi-Hamiltonian Systems Arising in the Problem of Integrable Geodesic Flows on the Two-Dimensional Torus

On Some Properties of Semi-Hamiltonian Systems Arising in the Problem of Integrable Geodesic Flows on the Two-Dimensional Torus

Mass spectrum in a six-dimensional SU(n) gauge theory on a magnetized torus

We examine six-dimensional SU(n) gauge theories compactified on a two-dimensional torus with a constant magnetic flux background to obtain a comprehensive low-energy mass spectrum. We introduce general background configurations including the magnetic flux and continuous Wilson line phases, consistent with classical equations of motion. Under the standard gauge fixing procedure, the complete mass spectrum in low-energy effective theory for the SU(n) case is newly presented without imposing restrictions on the gauge fixing parameter. Our analysis confirms the inevitable existence of tachyonic modes, which neither depend on the background configurations of Wilson line phases nor are affected by the gauge fixing parameter. Masses for some low-energy modes exhibit dependence on the gauge fixing parameter, and these modes are identified as would-be Goldstone bosons that are absorbed by massive four-dimensional vector fields. We discuss the phenomenological implications associated with stabilization or condensation of the tachyonic states. Various mass spectra and symmetry-breaking patterns are expected with flux backgrounds in the SU(n) case. They are helpful for constructing phenomenologically viable models beyond the standard model, such as gauge-Higgs unification and grand unified theories.

Local Well-Posedness of the Periodic Nonlinear Schrödinger Equation with a Quadratic Nonlinearity $$\overline{u}^2$$ in Negative Sobolev Spaces

AbstractWe study low regularity local well-posedness of the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $$\overline{u}^2$$ u ¯ 2 , posed on one-dimensional and two-dimensional tori. While the relevant bilinear estimate with respect to the $$X^{s, b}$$ X s , b -space is known to fail when the regularity s is below some threshold value, we establish local well-posedness for such low regularity by introducing modifications on the $$X^{s, b}$$ X s , b -space.