Platonic Hyperbolic Monopoles

We construct a number of explicit examples of hyperbolic monopoles, with various charges and often with some platonic symmetry. The fields are obtained from instanton data in ${\mathbb{R}^4}$ that are invariant under a circle action, and in most cases the monopole charge is equal to the instanton charge. A key ingredient is the identification of a new set of constraints on ADHM instanton data that are sufficient to ensure the circle invariance. Unlike for Euclidean monopoles, the formulae for the squared Higgs field magnitude in the examples we construct are rational functions of the coordinates. Using these formulae, we compute and illustrate the energy density of the monopoles. We also prove, for particular monopoles, that the number of zeros of the Higgs field is greater than the monopole charge, confirming numerical results established earlier for Euclidean monopoles. We also present some one-parameter families of monopoles analogous to known scattering events for Euclidean monopoles within the geodesic approximation.     

Pluricomplex Geometry and Hyperbolic Monopoles

In an attempt to combine non-commutative geometry and quantum gravity, Aastrup-Grimstrup-Nest construct a semi-finite spectral triple, modeling the space of G-connections for G = U(1) or SU (2). AGN show that the interaction between the algebra of holonomy loops ${\mathcal{B}}$ and the Dirac type operator ${\mathcal{D}}$ quantizes the Poisson structure of General Relativity in Ashtekar’s loop variables. This article generalizes AGN’s construction to any connected compact Lie group G. A construction of AGN’s semi-finite spectral triple in terms of an inductive limit of spectral triples is formulated. The refined construction permits the semi-finite spectral triple to be even when G is even dimensional. The Dirac-type operator ${\mathcal{D}}$ in AGN’s semi-finite spectral triple is a weighted sum of a basic Dirac operator on G. The weight assignment is a diverging sequence that governs the “volume” associated to each copy of G. The JLO cocycle of AGN’s triple is examined in terms of the weight assignment. An explicit condition on the weight assignment perturbations is given, so that the associated JLO class remains invariant. Such a condition leads to a functoriality property of AGN’s construction.     

Hyperbolic Calorons, Monopoles, and Instantons

We construct families of SO(3)-symmetric charge 1 instantons and calorons on the space . We show how the calorons include instantons and hyperbolic monopoles as limiting cases. We show how Euclidean calorons are the flat space limit of this family. Supported by PPARC.     

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.     

Monopoles and holography

Several lattice collaborations performing simulations with 2+1 light dynamical quarks have experienced difficulties in fitting their data with standard N _f = 3 chiral expansions at next-to-leading order, yielding low values of the quark condensate and/ or the decay constant in the N _f = 3 chiral limit. A reordering of these expansions seems required to analyse these data in a consistent way. We discuss such a reordering, known as Resummed Chiral Perturbation Theory, in the case of pseudoscalar masses and decay constants, pion and kaon electromagnetic form factors and K _ℓ3 form factors. We show that it provides a good fit of the recent results of two lattice collaborations (PACS-CS and RBC/UKQCD). We describe the emerging picture for the pattern of chiral symmetry breaking, marked by a strong dependence of the observables on the strange quark mass and thus a significant difference between chiral symmetry breaking in the N _f = 2 and N _f =3 chiral limits. We discuss the consequences for the ratio of decay constants F _K /F _π and the K ℓ3 form factor at vanishing momentum transfer.

     

Monopoles and Clusters

We define and study certain hyperkähler manifolds which capture the asymptotic behaviour of the SU(2)-monopole metric in regions where monopoles break down into monopoles of lower charges. The rate at which these new metrics approximate the monopole metric is exponential, as for the Gibbons-Manton metric.     

Monopoles and geodesics

Using the holomorphic geometry of the space of straight lines in Euclidean 3-space, it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the Atiyah-Ward Ansatz .     

Monopoles and Baker functions

The work in this paper pertains to the solutions of Nahm's equations, which arise in the Atiyah-Drinfield-Hitchin-Manin-Nahm construction of solutions to the Bogomol'nyi equations for static monopoles. This paper provides an explicit construction of the solution of Nahm's equations which satisfy regularity and reality conditions. The Lax form of Nahm's equations is reduced to a standard eigenvalue problem by a special gauge transformation. These equations may then be solved by the method of Baker-Krichever. This leads to a compact representation of the solutions of Nahm's equations. The regularity condition is shown to be related to the monodromy of the gauge reduced linear operator. Hitchin showed that the solutions of Nahm's equations can be characterized by an algebraic curve and some data on that curve. Here, this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential. Donaldson coordinatized the moduli space ofk-monopoles by a class of rational maps from the Riemann sphere to itself. The data of a Baker function is equivalent to this map. This method gives an apriori construction of the (known) two monopole solutions. We also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge. These solutions correspond to reducible curves with elliptic components and the associated Donaldson rational function has a simple partial fraction expansion.Supported in part by the National Science Foundation, Grant Number DMS-8701318 and the Arizona Center for Mathematical Sciences, sponsored by AFOSR Contract F49620-86-C0130 with the University Research Initiative Program at the University of Arizona     

Monopoles admit Fermi statistics

't Hooft-Polyakov monopoles are shown to admit Fermi statistics in the sense of Finkelstein.     

Perturbation of Singular Equilibria of Hyperbolic Two-Component Systems: A Universal Hydrodynamic Limit

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under the Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws:with where is a convex compact polygon in ². The system is typically strictly hyperbolic in the interior of with possible non-hyperbolic degeneracies on the boundary . We consider the case of an isolated singular (i.e. non-hyperbolic) point on the interior of one of the edges of , call it (₀,u₀). We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities (₀,u₀) of the conserved quantities. We prove that for a very rich class of systems, under a proper hydrodynamic limit the propagation of these small perturbations are universally driven by the two-by-two systemwhere the parameter is the only trace of the microscopic structure.The proof relies on the relative entropy method and thus, it is valid only in the regime of smooth solutions of the pde. But there are essential new elements: in order to control the fluctuations of the terms with Poissonian (rather than Gaussian) decay coming from the low density approximations we have to apply refined pde estimates. In particular Lax entropies of these pde systems play a not merely technical key role in the main part of the proof.     

Monopoles and their symmetries

Properties of spherically symmetric monopoles are discussed. Inversion symmetry is also considered.     

Monopoles and family-replicated unification

The present theory is based on the assumption that, at very small (Planck scale) distances our spacetime is discrete, and this discreteness influences the Planck scale physics. Considering our (3+1)-dimensional spacetime as a regular hypercubic lattice with a parameter a=λ_Pl, where λ_Pl is the Planck length, we have investigated a role of lattice artifact monopoles, which is essential near the Planck scale if the family-replicated gauge group model (FRGGM) is an extension of the Standard Model (SM) at high energies. It was shown that monopoles have N times smaller magnetic charge in the FRGGM than in the SM (N is the number of families in the FRGGM). These monopoles can give an additional contribution to β functions of the renormalization-group equations for the running fine structure constants α_i(μ) (i=1, 2, 3 correspond to the U(1), SU(2), and SU(3) gauge groups of the SM). We have used the Dirac relation for renormalized electric and magnetic charges. Also, we have estimated the enlargement of a number of fermions in the FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. The different role of monopoles in the vicinity of the Planck scale gives rise either to anti-GUT or to the new possibility of unification of gauge interactions (including gravity) at the scale μ_GUT≈10^18.4 GeV. We discussed the possibility of the [SU(5)]³ SUSY or [SO(10)]³ SUSY unifications.     

关于磁单极子的探索

磁单极子的概念自狄拉克提出以来,科学家一直都在努力寻找。虽然在理论上进行了深入的研究,但是在实验中,迄今仍然没能找到它们存在的确凿证据。文章从麦克斯韦方程组的对称性及电荷量子化的角度介绍了磁单极子的提出,磁单极子的单位及性质。并介绍了关于磁单极子的实验探索,除了历史上著名的两次实验外,科学家还在自旋冰中观察到磁单极子的“准粒子”。     

Monopoles,vortices and confinement

An exact relation is established between an SO(3) lattice gauge theory model without monopoles, and a corresponding SU(2) model. Elimination of the monopoles (and their strings) leads to a substantial lowering of the entropy of thin vortices and a corresponding decrease of the string tension for low β. This is revealed by approximate calculations of the vortex free energy and is confirmed by Monte Carlo data. The value of the physical transition temperature to “hot gluon soup” is also lowered considerably.