Gauge Potential Decomposition Research Articles

Decomposition theory of gauge potential and global topology problems

Using the idea of decomposability of gauge potential and internal structure put forward by Duan Yi-Shi, we decompose gauge potential of SO( n ) group in the unit vector field by using geometric algebra method, we get the general decomposition form and discuss the properties of this decomposition. In this paper, we give the form of decomposition of SU(2) group and U(1) group with unit vector field, which is exactly the result given by famous physicist Fadeev in 1999. The local topological structure of Gauss-Bonnet-Chern density is discussed by using the general form of decomposition of SO( n ) gauge potential. The global topological structure of the density is Gauss-Bonnet-Chern theorem, and the Morse theoretical form of Euler-Poincare characteristic is easily obtained from the topological structure. A new circulation condition is obtained by using the normal potential decomposition of SU(2) gauge potential to study –1/2 Bose-Einstein condensate, which is also a generalization of the Mernin-Ho relation. Finally, using the relation between Torsion tensor and U(1) gauge theory of three-dimensional Riemannian geometry discovered by Duan Yi-Shi, the relation between dislocation line and link number is studied by using U(1) gauge potential decomposition.

Gauge potential decomposition and topological current theories with applications in the study of topological insulators

The theories of gauge potential decomposition and topological currents proposed by Prof. YiShi Duan et al. have remarkable significance in the research of topological inner structures of gauge fields, with wide applications in the analysis of topological defects/excitations in various physical systems. In this paper, firstly, a brief survey for these theories is presented. Secondly, the theories are applied in the study of topological insulators in two dimensions. Topological insulators are at the cutting edge of current international research of condensed matter physics, where topological effects stem from the non-trivial topological numbers, such as the Chern numbers. In this paper different types of topological defects are investigated within one single framework, the so-called t Hooft monopole model, which acts as a $U(1)$ sub-field theory of a non-Abelian gauge field.

Topological structure of the solitons solution in SU(3) Dunne-Jackiw-Pi-Trugenberger model

By using φ-mapping topological current theory and gauge potential decomposition, we discuss the self-dual equation and its solution in the SU(N) Dunne-Jackiw-Pi-Trugenberger model and obtain a new concrete self-dual equation with a δ function. For the SU(3) case, we obtain a new self-duality solution and find the relationship between the soliton solution and topological number which is determined by the Hopf index and Brouwer degree of φ-mapping. In our solution, the flux of this soliton is naturally quantized.

The Generalization of Chern-Simons Current and the Topological Tensor Current of p-Branes

To make the gauge field theory foundation of the topological current of p-branes introduced in our previous work, we present a novel topological tensor current in SO(N) gauge field theory. This non-Abelian gauge field tensor current is the straightforward generalization of the Chern-Simons topological current of strings. By making use of the SO(N) gauge potential decomposition theory and the φ-mapping topological current theory, it is proved that the p-brane is created at every isolated zero of the Clifford vector field $\overrightarrow{\phi }(x)$ and the charges carried by p-branes are topologically quantized and labelled by the winding number of the φ-mapping.

Decomposition of Noncommutative U(1) Gauge Potential

We investigate the decomposition of noncommutative gauge potential Âi, and find that it has inner structure, namely, Âi can be decomposed in two parts, b̂i and âi, where b̂i satisfies gauge transformations while âi satisfies adjoint transformations, so dose the Seiherg–Witten mapping of noncommutative U(1) gauge potential. By means of Seiberg–Witten mapping, we construct a mapping of unit vector field between noncommutative space and ordinary space, and find the noncommutative U(1) gauge potential and its gauge field tensor can be expressed in terms of the unit vector field. When the unit vector field has no singularity point, noncommutative gauge potential and gauge field tensor will equal ordinary gauge potential and gauge field tensor

Self-dual Vortices in Schrödinger–Chern–Simons Model

By making use of the U(1) gauge potential decomposition theory and the ϕ-mapping topological current theory, we investigate the Schrödinger–Chern–Simons model in the thin-film superconductor system and obtain an exact Bogomolny self-dual equation with a topological term. It is revealed that there exist self-dual vortices in the system. We study the inner topological structure of the self-dual vortices and show that their topological charges are topologically quantized and labeled by Hopf indices and Brouwer degrees. Furthermore, the vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the vector field ϕ⃗.

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.

TOPOLOGICAL STRUCTURE OF THE MANY VORTICES SOLUTION IN JACKIW–PI MODEL

By using the gauge potential decomposition, we discuss the self-dual equation and its solution in Jackiw–Pi model. We obtain a new concrete self-dual equation and find relationship between Chern–Simons vortices solution and topological number which is determined by Hopf indices and Brouwer degrees of Ψ-mapping. To show the meaning of topological number we give several figures with different topological numbers. In order to investigate the topological properties of many vortices, we use five parameters (two positions, one scale, one phase per vortex and one charge of each vortex) to describe each vortex in many vortices solutions in Jackiw–Pi model. For many vortices, we give three figures with different topological numbers to show the effect of the charge on the many vortices solutions. We also study the quantization of flux of those vortices related to the topological numbers in this case.

TOPOLOGICAL ZERO-THICKNESS COSMIC STRINGS

In this paper, based on the gauge potential decomposition and the ϕ-mapping theories, we study the topological structures and properties of the cosmic strings that arise in the Abelian–Higgs gauge theory in the zero-thickness limit. After a detailed discussion, we conclude that the topological tensor current introduced in our model is a better and more basic starting point than the generally used Nambu–Goto effective action for studying cosmic strings.

Topological Structure of Vortex Solution in Jackiw–PiModel

By using ϕ-mapping method, we discuss the topologicalstructure of the self-duality solution in Jackiw–Pi model in termsof gauge potential decomposition. We set up relationship betweenChern–Simons vortex solution and topological number, which isdetermined by Hopf index and Brouwer degree. We also give thequantization of flux in this case. Then, we study the angularmomentum of the vortex, which can be expressed in terms of the flux.

Topological Solitons in the CP N Model

We investigate the solitons in the CPN 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.

TOPOLOGICAL STRUCTURE AND EVOLUTION OF SPACE–TIME DISLOCATIONS AND DISCLINATIONS

By making use of the gauge potential decomposition theory and ϕ-mapping theory, the topological structure and the topological quantization of dislocations and disclinations are studied in the framework of Riemann–Cartan space–time manifold. The evolution of dislocation strings and disclination points is also studied from the topological properties of the order parameter field. The dislocations and disclinations are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of the order parameter field.

Many knots in Chern-Simons field theory

In this paper, using the $U(1)$ gauge potential decomposition and the $\ensuremath{\varphi}$-mapping topological current theory, the many knotlike vortex lines in Chern-Simons field theory are studied. It has been pointed out that the Chern-Simons action is a topological invariant for the family of knots; i.e., it is just the total sum of all the self-linking and all the linking numbers of the knot family. Furthermore, it is also shown that this Chern-Simons knot topological number is preserved in the branch processes (splitting, merging, and intersection) during the evolution of these knotlike vortex lines.

Decomposition theory of the U(1)gauge potential and the London assumption in topological quantum mechanics

The decomposition theory of the U(1)gauge potential has been studied. A rigorous proof of the London assumption isgiven. By making use of this gauge potential decomposition and the ϕ-mappingtopological current theory, the precise expression for ∇⃗×V⃗ is obtained, and the topologyin quantum mechanics is discussed.

Topological Structure of Chern–Simons Vortex**The project supported by National Natural Science Foundation of China

Based on the gauge potential decomposition theory and the -mapping method, the topological inner structure of the Chern–Sirnons–Higgs vortex has been studied strictly. It is shown that there exits a multi-charged vortex at every zero point of the Higgs scalar field The multivortex solutions in the Chern–Simons–Higgs model are obtained strictly.

Evolution of the Chern-Simons vortices

Based on the gauge potential decomposition theory and the $\ensuremath{\varphi}$-mapping theory, the topological inner structure of the Chern-Simons-Higgs (CSH) vortex is discussed in detail. The evolution of CSH vortices is also studied from the topological properties of the Higgs scalar field. The vortices are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of the scalar field \ensuremath{\varphi}.

THE QUANTIZATION AND PRODUCTION OF THE SPACE–TIME DEFECTS IN THE EARLY UNIVERSE

In Riemann–Cartan manifold U4, a new topological invariant is obtained by means of the torsion tensor. In order to describe the space–time defects (which appear in the early universe due to torsion) in an invariant form, the new topological invariant is introduced to measure the size of defects and it is interpreted as the dislocation flux in internal space. By the use of the so-called ϕ mapping method and the gauge potential decomposition, the dislocation flux is quantized in units of the Planck length. The quantum numbers are determined by the Hopf indices and the Brouwer degrees. Furthermore, the dynamic form of the dislocations is studied by defining an identically conserved current. Based on the implicit function theorem, the production of the defects at the limit points is detailed.

Gauge potential decomposition, space-time defects, and Planck’s constant

A new geometrization of Planck’s constant is proposed, which also bases on the torsion in the space-time. A quantity is introduced to describe the space-time dislocations that appear due to torsion. It will be pointed out that there is U(1)-like gauge invariance in this new geometrization. Using the gauge-potential decomposition, the quantity is quantized in units of the Planck length. The quantum number is determined by Hopf indices and Brouwer degrees.

Topological gauge theory of dislocation in general decomposition gauge potential formulation

We detail the general decomposition of gauge potential by a unit vector field which is not necessarily being gauge parallel and consequently present the fundamental formulas for discussing the topological properties of dislocations in the topological gauge field theory of dislocation and disclination continuum.

Topological current structure of disclinations in the 4-dimensional gauge field theory of dislocation and disclination continuum

The topological current structure of disclinations in the 4-dimensional gauge field theory of dislocation and disclination continuum is approached by making use of the decomposition of gauge potential. It is shown that the topological stable disclinations are identified with the disclination points, which are classified in terms of the topological invariant-wrapping number. The topological current of the disclinations is derived to correspond to a 4-dimensional kinetics current of a set of moving point-like objects which is just the zeros of the parameters field. The velocity of the topological disclination is determined by the Jacobians of the parameters field at the zero points. It is also shown that the disclination points are characterized by Brouwer degree and Hopf index.