Singular Hyperbolic Monopoles

We study the twistor theory of singular hyperbolic SU(2) monopoles following the approach taken by Kronheimer [9] in the Euclidean case. We use our results to show that the moduli space of charge 1 monopoles possesses a natural 2-sphere of scalar flat Kähler metrics. In the zero mass limit, the metrics reduce to a class of metrics first studied by LeBrun in [10].     

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.     

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.     

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.

     

Canonical Sasakian Metrics

Let M be a closed manifold of Sasaki type. A polarization of M is defined by a Reeb vector field, and for any such polarization, we consider the set of all Sasakian metrics compatible with it. On this space we study the functional given by the square of the L ²-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open. We use our results to describe fully the case of the sphere with its standard CR structure, showing that each element of its Sasaki cone can be represented by a canonical metric; we compute their Sasaki-Futaki invariant, and use it to describe the canonical metrics that have constant scalar curvature, and to prove that only the standard polarization can be represented by a Sasaki-Einstein metric. During the preparation of this work, the first two authors were partially supported by NSF grant DMS-0504367.     

Extremal Sasakian horizons

We point out a simple construction of an infinite class of Einstein near-horizon geometries in all odd dimensions greater than five. Cross-sections of the horizons are inhomogeneous Sasakian metrics (but not Einstein) on S³×S²$S^3\times S^2$ and more generally on S³$S^3$-bundles over any compact positive Kähler–Einstein manifold. They are all consistent with the known topology and symmetry constraints for asymptotically flat or globally AdS black holes.     

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 .     

Some Comments on Quantum Magnetic Monopoles

In this paper we intend to present some path-integral studies in the problem of confinement in the presence of fermionic and scalar magnetic monopole fields through:

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.     

On Eta-Einstein Sasakian Geometry

A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kähler orbifold . In the case when the transverse space is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.     

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.     

Lagrangian Curves on Spectral Curves of Monopoles

We study Lagrangian points on smooth holomorphic curves in ${\rm T}{\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of ${\rm T}{\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$ , these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$ , which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$ , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.     

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.     

Monopoles,duality, triality

The existence of magnetic charges could be a raison d'être not only for the quantization of electricity in units $\text{1}\text{3}\text{e}$ but also for the confinement of the quarks at the end of “observable”, “electric” Dirac strings (quarks have no magnetic charge in this scheme). We first review the Dirac quantization condition, using a “sum over histories” approach, and get a more general result: A string attached to the dyon (e₁, g₂) is observable by a dyon (e₂, g₂) unless (1?x) e₂g₁ ? xe₁g₂ = nh, where x is an arbitrary parameter which reflects an ambiguity in the action principle. The Dirac and Schwinger-Zwanziger quantization rules are special cases, with $\text{x = 0}\text{and}\text{1}\text{2}$, respectively. Then, we look for the values of x and of the magnetic charges to “explain” that (i) the electric charge is quantized in units $\text{1}\text{3}\text{e}$, (ii) the string attached to an electron is unobservable, (iii) the string attached to a quark is observable. We find a denumerable set of solutions. In most cases, the magnetic charges also are connected with observable strings.