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In the light of -mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of . 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 strengths. From the Lagrange's method of multipliers, the equilibrium equation and the molecular field of liquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.

In this paper, the decomposition of SU(2) gauge potential in terms of Pauli spinor is studied. Using this decomposition, the spinor structures of Chern-Simons form and the Chern density are obtained. Furthermore, the knot quantum number of non-Abelian gauge theory can be expressed by the Chern-Simons spinor structure, and the second Chern number is characterized by the Hopf indices and the Brouwer degrees of ?-mapping.

Using -mapping method and topological current theory, the topological structure and bifurcation of disclination lines in two-dimensional liquid crystals are studied. By introducing the strength density and the topological current of many disclination lines, the total disclination strength is topologically quantized by the Hopf indices and Brouwer degrees at the singularities of the director field when the Jacobian determinant of director field does not vanish. When the Jacobian determinant vanishes, the origin, annihilation and bifurcation processes of disclination lines are studied in the neighborhoods of the limit points and bifurcation points, respectively. The branch solutions at the limit point and the different directions of all branch curves at the bifurcation point are calculated with the conservation law of the topological quantum numbers. It is pointed out that a disclination line with a higher strength is unstable and it will evolve to the lower strength state through the bifurcation process.

In this paper, the author gives a brief survey of the gauge theories of gravitation and presents recent developments on this aspect.

In this presentation, the quantization and renormalization of the Yang-Mills field theory are discussed from the standpoint of the decomposition of gauge potential. It may be regarded as a new basis for the formulation of the fundamental interactions of particle theories.

By decomposing the Bogomol'nyi self-dual equation in the Abelian Chern–Simons Higgs model, we find a self-dual topological term that was ignored all the time in the Bogomol'nyi self-duality equation due to the improper decomposition of the complex Higgs field. We also present a new self-dual equation that includes the topological term. It is shown that the self-dual vortex just arises from the symmetric phase of the Higgs field φ = 0. Using our φ-mapping theory, the inner topological structure of the vortex and double vortex is given.

By the method of -mapping topological current theory, the bifurcation behavior of the topological current is discussed in detail in the symmetrical time-dependent Ginzburg-Landau model at the critical points of the order parameter field. The different directions of the branch curves at the critical point have been obtained.