The Topological Quantization and Bifurcation of strings in Early Universe

We present a new topological invariant to describe the spacetime defect in Riemann- Cartan manifold. By means of the φ-mapping theory and the rank-two topological tensor current (associated to the new topological invariant), we show that there must exist many string objects which naturally generated from the zero points of φ-mapping and is quantized in the topological level under the condition that the Jacobian J(φ/ν) ≠ 0. When J (φ/ν) = 0, we find that there exists a crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the strings are detailed in the neighborhoods of the limit points and bifurcation points of φ-mapping, respectively.