Stability of Disclinations in Nematic Liquid Crystals

In the light of C-mapping method and topological current theory, the stability of disclinations around a spherical particle in nematic liquid crystals is studied. We consider two different defect structures around a spherical particle： disclination ring and point defect at the north or south pole of the particle. We calculate the free energy of these different defects in the elastic theory. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclinations. The other is the free energy density of disclinations themselves, which is shown to be concentrated at the defect and to be topologically quantized in the unit of （k - k24）π/2. It is shown that in the presence of saddle-splay elasticity a dipole （radial and hyperbolic hedgehog） configuration that accompanies a particle with strong homeotropic anchoring takes the structure of a small disclination ring, not a point defect.

Topological Defects in Liquid Crystal Films

A topological theory of liquid crystal films in the presence of defects is developed based on the Ф-mapping topological current theory. By generalizing the free-energy density in ＂one-constant＂ approximation, a covariant free- energy density is obtained, from which the U（1） gauge field and the unified topological current for monopoles and strings in liquid crystals are derived. The inner topological structure of these topological defects is characterized by the winding numbers of Ф-mapping.

Effect of Disclination Lines on Free Energy of Nematic Liquid Crystals

In the light of φ-mapping method-and topological current theory, the effect of disclination lines on the free energy density of nematic liquid crystals is studied. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclination lines. The other is the saddle-splay energy density, which is shown to be centralized at the disclination lines and to he topologically quantized in the unit of kπ/2 when the Jacobian determinant of the director field does not vanish at the singularities of the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed in the neighborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination line with high topological quantum number is unstable and will evolve to the low topological quantum number states through the splitting process.

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.

Topological Structure of Phase Vortex in Resonating Valence Bond Superconductivity

In this paper, based on the Schrfdinger equation and the ψ mapping theory, the accurate expression for the gradient of resonating valence bond superconducting phase Θ^2s is found. The expression of Δ↓Θ^2s is just the velocity flow without considering the coefficient. The curl of Δ↓Θ^2s is where the vortex lies, and has important relation to δ2（ψ） and an important relation to the zero points of resonating valence bond superconducting order parameter ψ. The topological structure of the vortex is characterized by the ψ-mapping topological numbers Hopf-index and Brouwer degrees. The Ginzberg-Landau equation in resonating valence bond state also is discussed in this theory. The magnetic property is discussed also.

Topological Structure and Topological Tensor Current of Gauss-Bonnet-Chern Theorem

We offer an intrinsic theoretical framework to reveal the inner relationships among three theories for Euler characteristic number, including Gauss Bonnet-Chern theorem, Hop-Poincaré theorem and Morse theory. Moreover, we consider the Gauss Bonnet-Chern （GBC） form imbedded in arbitrary higher-dimensional manifold, which suggests a Hodge dual tensor current. We show the brane structure inherent in the GBC tensor current and obtain the generalized Nambu action for the multi branes with quantized topological charge.

Fermionic Zero Modes in Self-dual Vortex Background on a Torus

We study fermionic zero modes in the self-dual vortex background on an extra two-dimensional Riemann surface in （5＋1） dimensions. Using the generalized Abelian-Higgs model, we obtain the inner topological structure of the self-dual vortex and establish the exact self-duality equation with topological term. Then we analyze the Dirac operator on an extra torus and the effective Lagrangian of four-dimensional fermions with the self-dual vortex background. Solving the Dirac equation, the fermionic zero modes on a torus with the self-dual vortex background in two simple cases are obtained.

Surface Distribution of Topological Defects on Axisymmetric

We propose a general method of deterrnining the distribution of topological defects on axisymmetric surface, and study the distribution of topological defects on biconcave-discoid surface, which is the geometric configuration of red blood cell. There are three most possible cases of the distribution of the topological defects on the biconcave surface： four defects charged with 1/2, two defects charged with ＋1, or one defect charged with 2. For the four defect charged with 1/2, they sit at the vertices of a square imbedded in the equator of biconcave surface.

Multi-types of Skyrmions in SU（N） Quantum Hall System

The skyrmions in SU（N） quantum Hall （QH） system are discussed. By analyzing the gauge field structure and the topological properties of this QH system it is pointed out that in the SU（N） QH system there can exist （N-1） types of skyrmion structures, instead of only one type of skyrmions. In this paper, by means of the Abelian projections according to the （N - 1） Cartan subalgebra local bases, we obtain the （N - 1） U（1） electromagnetic field tensors in the SU（N） gauge field of the QH system, and then derive （N - 1） types of skyrmion structures from these U（1） sub-field tensors. Furthermore, in light of the C-mapping topological current method, the topological charges and the motion of these skyrmions are also discussed.

Integer and Half-Integer Quantization Conditions in Quantum Mechanics

The integer and half-integer quantization conditions are found in quantum mechanics based on the topological structure of symmetry group of the singlet and spinor wavefunction.The internal symmetry of physical system is shown to be sufficient to determine the topological structure in quantum mechanics without taking into account the dynamical details about the interaction.

Topological Quantization of k-Dimensional Topological Defects and Motion Equations

UsingФ-mapping method and kth-order topological tensor current theory,we present a unified theory of describing k-dimensional topological defects and obtain their topological quantization and motion equations.It is shown that the inner structure of the topological tensor current is just the dynamic form of the topological defects,which are generated from the zeros of the m-component order parameter vector field.In this dynamic form,the topological defects are topologically quantized naturally and the topological quantum numbers are determined by the Hopf indices and the Brouwer degrees.As the generalization of Nielsen's Lagrangian and Nambu's action for strings,the action and the motion equations of the topological defects are also derived.

Evolution of the Topological Linear Defects

In the light of ø-mapping method and topological current theory,the topological structure and the topological quantization of topological linear defects are obtained under the condition that Jacobian J(ø/v)≠0.When J(ø/v)=0,it is shown that there exists the crucial case of branch process.Based on the implicit function theorem and the Taylor expansion,the generation,annihilation and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of ø-mapping,respectively.

Topological Quantization of Linear Defects

Using Ф-mapping method and topological current theory,we get the topological structure and the topological quantization of topological linear defects and point out that the topological quantum numbers of the linear defects are described by the Winding numbers of Ф-mapping which are determined in terms of the Hopf indices and the Brouwer degrees.All the topological linear defects are generated from the zero points of the Ф-mapping.

Reply to the Comment by ZHAO

Recently,ZHAO made a comment saying that the result in Ref.1 as well as the whole theory of Ref.2 are wrong based on his unacceptable results.We point out here that the unacceptable results in the comment were founded on the neglect of the boundary conditions of vierbein and Lorentz transforms.

About the Energy of the Universe

We use the general covariant conservation law of energy-momentum,which is general covariant and therefore preferable to the non-covariant conservation laws,to analyse the energy of the universe and obtained vanishing energy.A physical interpretation of the vanishing energy and some discussions are also presented.

Fermionic Zero Modes in Gauge and Gravity Backgrounds on a Sphere

In this letter we study fermionic zero modes in gauge and gravity backgrounds taking a two-dimensional compact manifold S2 as extra dimensions. The result is that there exist massless Dirac fermions which have normalizable zero modes under quite general assumptions about these backgrounds on the bulk. Several special cases of gauge background on the sphere axe discussed and some simple fermionic zero modes are obtained.

Self-dual Vortices in Schrdinger-Chern-Simons Model

By making use of the U（1） gauge potential decomposition theory and the Ф-mapping topological current theory, we investigate the Schroedinger-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 Ф.