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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.

Using φ-mapping method and topological current theory, the properties and behaviors of disclination points in three-dimensional liquid crystals are studied. By introducing the strength density and the topological current of many disclination points, the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the general director field when the Jacobian determinant of the general director field does not vanish. When the Jacobian determinant vanishes, the origin, annihilation, and bifurcation of disclination points are detailed in the neighborhoods of the limit point and bifurcation point, respectively. The branch solutions at the limit point and the different directions of all branch curves at the first- and second-order degenerated points are calculated. It is pointed out that a disclination point with a higher strength is unstable and will evolve to the lower strength state through the bifurcation process. An original disclination point can split into at most four disclination points at one time.

We construct an elementary topological current in composite-boson field andpoint out that the time component of this topological current is the density ofsolitons in the field. Based on the implicit function theorem and the Taylorexpansion, the evolution of solitons is detailed in the neighborhoods of thebranch points of the composite-boson field. We also find that solitons generateor annihilate at the limit points and encounter one another, split, or merge at thebifurcation points of the composite-bosons field.

We present a new topological invariant todescribe space-time defects which is closely related tothe torsion tensor in a Riemann–Cartan manifold.By virtue of the topological current theory andφ-mapping method, we show that there must existmultistring objects generated from the zero points ofthe φ-mapping. These strings are topologicallyquantized. The topological quantum numbers are thewinding numbers described by the Hopf indices and the Brouwerdegrees of the φ-mapping.

Using SU(2) gauge field theory and theφ-mapping method, we quantize the magnetic monopolesat the topological level and determine their quantumnumbers by the Hopf indices and Brouwer degrees of the φ-mapping. Then, based on the implicitfunction theorem and Taylor expansion, we study theorigin and bifurcation theories of magnetic monopoles atthe limit points and bifurcation points (includingfirst-order and second-order degenerate points),respectively. We point out that a magnetic monopole cansplit into at most four particles at one time.

In a Riemann–Cartan manifold a topologicalinvariant is constructed in terms of the torsion tensor.Using the φ-mapping method and the completedecomposition of the gauge potential, the topologicalinvariant is extricated from a strong restrictivecondition and is quantized in units of an elementarylength. This topological invariant is linked to thefirst Chern class and its inner structure is labeled bya set of winding numbers. In the early universe,by extending to a gauge parallel basis in internal spaceand four analogous topological invariants, thespace-time defects are formulated in an invariant form and are quantized naturally in units of thePlanck length.

In the early universe, a new topological invariant is interpreted as the space-time dislocation flux and is quantized in the topological level. By extending to a topological current of dislocations, the dynamic form of the defects is obtained under the condition that the Jacobian determinant D(φ/u) ≠ 0. When D(φ/u) = 0, it is shown that there exists the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the space-time dislocations are detailed in the neighborhoods of the limit points and bifurcation points of φ-mapping, respectively. It is pointed out that, since the dislocation current is identically conserved, the total topological quantum numbers of the branched dislocation fluxes will remain constant during their origin and bifurcation processes, which are important in the early universe because of spontaneous symmetry breaking.

A general covariant conservation law of energy-momentum in complex general relativity is obtained by way of general displacement transformation in terms of Ashtekar's new variables. The energy is exactly the adm Hamiltonian on the constraint surface on condition that an appropriate time function is chosen. The energy-momentum is gauge covariant and commutes with all the constraints whence they are physical observables. Furthermore, the Poisson brackets of the momentum and the internalSU(2) charges form a 3-Poincare algebra.

We present three operators in quantum mechanics that obey the commutation relations of quantum groupSUq(2). These operators are nonlinear combinations of the conventional angular momentum operators and are called the quantumq-analog angular momentum operators. When the quantum deformation parameterr = Inq vanishes, these quantumq-analog angular momentum operators reduce to the usual angular momentum operators.

In this paper, a new coordinate condition in general relativity is proposed. This coordinate condition is just Euler's equations of the harmonic map theory. It is shown that in case of rectangular coordinates our coordinate condition reduces to Fock's harmonic coordinate condition. Using this new coordinate condition the different spherical symmetric solutions of the Einstein equations are discussed. The axisymmetric case is also investigated.

We study the wave dislocations with an induced gauge potential. The topological current characterized the wave dislocations is constructed with the dual of Abelian gauge field. And the topological charges and locations of the wave dislocations are determined by the φmapping topological current theory. Furthermore, it is shown that the knotted wave dislocations can be described with a Hopf invariant in the wave field. At last we discussed the evolution of the knotted wave dislocations.

In the literatures, several types of thick smooth brane configurations in a pure geometric Weyl integrable 5-dimensional space time have been presented. The Weyl geometry is a non-Riemannian modification of 5-dimensional Kaluza--Klein (KK) theory. All these thick brane solutions preserve 4-dimensional Poincar�� invariance, and some of them break $Z_2$--symmetry along the extra dimension. In this paper, we study localization of various matter fields on these pure geometrical thick branes, which also localize the graviton. We present the shape of the potential of the corresponding Schr$\mathrm{\ddot{o}}$dinger problem and obtain the lowest KK mode. It is showed that, for both spin 0 scalars and spin 1 vectors, there exists a continuum gapless spectrum of KK states with $m^2>0$. But only the massless mode of scalars is found to be normalizable on the brane. However, for the massless left or right chiral fermion localization, there must be some kind of Yukawa coupling. For a special coupling, there exist a series of discrete massive KK modes with $m^2 >0$. It is also showed that for a given coupling constant only one of the massless chiral modes is localized on the branes., 17 pages, 5 figures, accepted by JHEP

We study localization of bulk fermions on a string-like defect with the exponentially decreasing warp factor in six dimensions with inclusion of U(1) gauge background from the viewpoint of field theory, and give the conditions under which localized spin 1/2 and 3/2 fermions can be obtained., Comment: Latex, 13 pages; V2: two references added; v3: 14 pages with some changes; v4: two references (wrongly deleted in the 3nd version) added Latex, 14 pages, accepted by JHEP; V5: published version

We discuss the conservation laws in general spacetime with torsion and nonmetricity by use of the Noether theorem. We give a general Lagrangian density, which can be reduced to the one given by Rauch and Nieh (1981). The general covariant energy-momentum conservation law is obtained with respect to the general displacement transformationX μ' =X μ +e μ b b , which is a special case of general conservation laws. It is shown that in such a case, the existence of superpotentials is assured. The results are the natural extension of references [1,2].