Quantized Lax Equations and Their Solutions

The O(3) sigma model and abelian Higgs model in two space dimensions admit topological (Bogomol'nyi) lower bounds on their energy. This paper proposes lattice versions of these systems which maintain the Bogomol'nyi bounds. One consequence is that instantons/solitons/vortices on the lattice then have a high degree of stability. Received: 29 February 1996 / Accepted: 5 August 1996     

Topological Aspects of Superconductors at Dual Point

We study the properties of the Ginzburg-Landau model at the dual point for the superconductors. By making use of the U（1） gauge potential decomposition and the φ-mapping theory, we investigate the topological inner structure of the Bogomol＇nyi equations and deduce a modified decoupled Bogomol＇nyi equation with a nontrivial topological term, which is ignored in conventional model. We find that the nontrivial topological term is closely related to the N-vortex, which arises from the zero points of the complex scalar field, Furthermore, we establish a relationship between Ginzburg Landau free energy and the winding number.     

Branes and Calibrated Geometries

The fivebrane worldvolume theory in eleven dimensions is known to contain BPS threebrane solitons which can also be interpreted as a fivebrane whose worldvolume is wrapped around a Riemann surface. By considering configurations of intersecting fivebranes and hence intersecting threebrane solitons, we determine the Bogomol'nyi equations for more general BPS configurations. We obtain differential equations, generalising Cauchy–Riemann equations, which imply that the worldvolume of the fivebrane is wrapped around a calibrated submanifold. Received: 20 April 1998 / Accepted: 16 November 1998     

Electric monopoles in topological field theories

A tensor product generalization ofB ∧F theories is proposed that has a Bogomol'nyi structure. Nonsingular, stable, finite-energy particle-like solutions to the Bogomol'nyi equations are studied. Unlike Yang-Mills(-Higgs) theory, the Bogomol'nyi structure does not appear as a perfect square in the Lagrangian. Consequently, the Bogomol'nyi energy can be obtained in more than one way. The added flexibility permits electric monopole solutions.     

Model for Topological Fermions

 We introduce a model designed to describe charged particles as stable topological solitons of a field with values on the internal space S ³. These solitons behave like particles with relativistic properties like Lorentz contraction and velocity dependence of mass. This mass is defined by the energy of the soliton. In this sense this model is a generalization of the Sine-Gordon model¹(We do not chase the aim to give a four-dimensional generalization of Coleman’s isomorphism between the Sine-Gordon model and the Thirring model which was shown in 2-dimensional space-time) from 1 + 1-dimensions to 3 + 1-dimensions, from S ¹ to S ³. For large distances from the centre of solitons this model tends to a dual U(1)-theory with freely propagating electromagnetic waves. Already at the classical level it describes important effects, which usually have to be explained by quantum field theory, like particle-antiparticle annihilation and the running of the coupling. Received November 30, 1999; revised June 20, 2000; accepted for publication October 2, 2000     

Dyons in topological field theories

We examine a class of topological field theories defined by Lagrangians that under certain conditions can be written as the sum of two characteristic numbers or winding numbers. Therefore, the action or the energy is a topological invariant and stable under perturbations. The sufficient conditions required for stability take the form of first-order field equations, analogous to the self-duality and Bogomol'nyi equations in Yang-Mills(-Higgs) theory. Solutions to the first-order equations automatically satisfy the full field equations. We show the existence of nontrivial, nonsingular, minimum energy spherically symmetric dyon solutions and that they are stable. We also discuss evidence for a dual field theory to Yang-Mills-Higgs in topological field theory. The existence of dual field theories and electric monopoles is predicted by Montonen and Olive.     

Topological Solitons in the <Emphasis Type="Italic">CP</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> Model

We investigate the solitons in the CP ^N model in terms of the decomposition of gauge potential. Based on the φ-mapping topological current theory, the charge and position of solitons is determined by the properties of the typical component. Furthermore, the motion and the bifurcation of multi-soliton is discussed. And the knotted solitons in high dimension is explored also.     

Scalar Gravity and Higgs Mechanism

The role that the auxiliary scalar field φ plays in Brans–Dicke cosmology is discussed. If a constant vacuum energy is assumed to be the origin of dark energy, then the corresponding density parameter would be a quantity varying with φ; and almost all of the fundamental components of our universe can be unified into the dynamical equation for φ. As a generalization of Brans–Dicke theory, we propose a new gravity theory with a complex scalar field ϕ which is coupled to the cosmological curvature scalar. Through such a coupling, the Higgs mechanism is naturally incorporated into the evolution of the universe, and a running density of the field vacuum energy is obtained which may release the particle standard model from the rigorous cosmological constant problem in some sense. Our model predicts a running mass scale of the fundamental particles in which the gauge symmetry breaks spontaneously. The running speed of the mass scale in our case could survive all existing experiments.     

Localized axion photon states in a strong magnetic field

We consider the axion field and electromagnetic waves with rapid time dependence, coupled to a strong time independent, asymptotically approaching a constant at infinity “mean” magnetic field, which takes into account the back reaction from the axion field and electromagnetic waves with rapid time dependence in a time averaged way. The direction of the self consistent mean field is orthogonal to the common direction of propagation of the axion and electromagnetic waves with rapid time dependence and parallel to the polarization of these electromagnetic waves. Then, there is an effective U(1) symmetry mixing axions and photons. Using the natural complex variables that this U(1) symmetry suggests we find localized planar soliton solutions. These solutions appear to be stable since they produce a different magnetic flux than the state with only a constant magnetic field, which we take as our “ground state”. The solitons also have non-trivial U(1) charge defined before, different from the uncharged vacuum. These solitons represent a new, non-gravitational mechanism, of trapping light. They could also affect the vacuum structure in models of the QCD vacuum that incorporate a magnetic condensate, introducing may be gluon axion solitons.     

Quasi-two-dimensional dark spatial solitons and generation of mixed phase dislocations

This paper presents experimental evidence that orthogonally crossed dark soliton stripes form quasi-two-dimensional spatial solitons with a soliton constant equal to that of singly charged optical vortices. Besides the pairs of oppositely charged optical vortex solitons, the snake instability of the dark formation at moderate saturation is found to lead to generation of steering mixed edge–screw phase dislocations with zero total topological charges. Received: 26 October 1998 / Revised version: 19 January 1999 / Published online: 12 May 1999     

Hopf solitons and Hopf Q-balls on S3

Field theories with a S²-valued unit vector field living on S³×ℝ space-time are investigated. The corresponding eikonal equation, which is known to provide an integrable sector for various sigma models in different spaces, is solved giving static as well as time-dependent multiply knotted configurations on S³ with arbitrary values of the Hopf index. Using these results, we then find a set of hopfions with topological charge Q_H=m², m∈Z, in the integrable subsector of the pure CP¹ model. In addition, we show that the CP¹ model with a potential term provides time-dependent solitons. In the case of the so-called “new baby Skyrme” potential we find, e.g., exact stationary hopfions, i.e., topological Q-balls. Our results further enable us to construct exact static and stationary Hopf solitons in the Faddeev–Niemi model with or without the new baby Skyrme potential. Generalizations for a large class of models are also discussed.     

Intriguing properties of kinks in a simple model with a two-component field

We present a summary of the topological and non-topological solitons of a two component field in 1+1 dimensions with application in field theory and in condensed matter physics. We note several intriguing analytical and numerical relationships between these solutions, which we believe to suggest that the relevant coupled nonlinear differential equations may be integrable exactly.     

What types of spatial soliton can be formed based on two-photon-isomerisation?

In this paper we answer the question: “what types of spatial soliton can be formed based on two-photon-isomerisation (TPI)”. The conclusion that anti-dark solitons are not supported by monotonic nonlinearity should cast light on the notion of fundamental spatial solitons. The idea to obtain a bright TPI soliton with the joining of a background light or with the coupling of a dark soliton offers new schemes of light-controlling-light.     

Discreteness effects on non-topological kink soliton dynamics in nonlinear lattices

We show that discreteness effects which are known to exist for topological solitons exist also for non-topological kink solitons in nonlinear lattices. Extending the technique previously proposed for topological kinks we exhibit three cases where the properties of narrow kinks in nonlinear lattices are qualitatively different. Supersonic solitons in a monoatomic chain can propagate at constant speed without losing energy due to discreteness. Subsonic kinks in a monoatomic chain permanently radiate small amplitude oscillations. In a diatomic chain both supersonic and subsonic kinks lose energy due to discreteness. The characteristics of the small amplitude oscillations radiated by the kinks, when they exist, are well determined by our theoretical approach. Additional weak nonlinear effects are also described and discussed.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

Polarization direction characters of local electric field around dielectric coated gold nanowire

The local electric field components in the dielectric wall with a long gold nanowire in its core are calculated based on quasi-static theory. The calculated results show that the complete polarized incident light does not only stimulate same directional complete polarized local electric field. The same directional polarized electric field only locates close to the poles of the core wire and is parallel or perpendicular to the polarized direction of the incident radiation. On the other hand, incident light also stimulates perpendicular directional polarization, which densely locates close to the poles of the core wire in the direction with an included angle π/4 or 3π/4 makes with polarization direction of incident light. Furthermore, local electric field components in the wall also depend on the dielectric constant of dielectric wall and surrounding medium. When dielectric constant of the wall is less than that of surrounding, the areas of perpendicular directional polarized local field in the wall reduce and shift greatly. At the same time, more parallel directional polarized local field focus in the poles of the wall along the incident polarization. PACS 78.67.Bf; 73.20.Mf; 36.40.Gk; 78.66.Bz; 73.20.Mf     

Balanced Topological Field Theories

We describe a class of topological field theories called “balanced topological field theories”. These theories are associated to moduli problems with vanishing virtual dimension and calculate the Euler character of various moduli spaces. We show that these theories are closely related to the geometry and equivariant cohomology of “iterated superspaces” that carry two differentials. We find the most general action for these theories, which turns out to define Morse theory on field space. We illustrate the constructions with numerous examples. Finally, we relate these theories to topological sigma-models twisted using an isometry of the target space. Received: 2 September 1996 / Accepted: 3 October 1996     

Solitonic fullerene structures in light atomic nuclei

The Skyrme model is a classical field theory which has topological soliton solutions. These solitons are candidates for describing nuclei, with an identification between the numbers of solitons and nucleons. We have computed numerically, using two different minimization algorithms, minimum energy configurations for up to 22 solitons. We find, remarkably, that the solutions for seven or more solitons have nucleon density isosurfaces in the form of polyhedra made of hexagons and pentagons. Precisely these structures arise, though at the much larger molecular scale, in the chemistry of carbon shells, where they are known as fullerenes.     

Semi-local cosmic strings and the cosmological constant problem

We study the cosmological constant problem in a three-dimensional N = 2 supergravity theory with gauge groupSU (2)_global × U(1)_local. The model we consider is known to admit string-like configurations, the so-called semi-local cosmic strings. We show that the stability of these solitonic solutions is provided by supersymmetry through the existence of a lower bound for the energy, even though the manifold of the Higgs vacuum does not contain non-contractible loops. Charged Killing spinors do exist over configurations that saturate the Bogomol'nyi bound, as a consequence of an Aharonov-Bohm-like effect. Nevertheless, there are no physical fermionic zero modes on these backgrounds. The exact vanishing of the cosmological constant does not imply, then, Bose-Fermi degeneracy. This provides a non-trivial example of the recent claim made by Witten on the vanishing of the cosmological constant in three dimensions without unphysical degeneracies.