The General Decomposition Theory of SU(2) Gauge Potential, Topological Structure and Bifurcation of SU(2) Chern Density

TL;DR

This paper develops a comprehensive geometric algebra-based decomposition of SU(2) gauge potentials on 4D manifolds, analyzing the topological structure and bifurcation behavior of the SU(2) Chern density, revealing its dependence on zero points of vector fields.

Contribution

It introduces a general decomposition theory for SU(2) gauge potentials and explores the topological and bifurcation properties of the SU(2) Chern density in detail.

Findings

Chern density can be expressed using delta functions at zero points of vector fields.

Zero points of vector fields are crucial for the topological properties of the manifold.

Bifurcation analysis shows the topological charge remains invariant during branch processes.

Abstract

By means of the geometric algebra the general decomposition of SU(2) gauge potential on the sphere bundle of a compact and oriented 4-dimensional manifold is given. Using this decomposition theory the SU(2) Chern density has been studied in detail. It shows that the SU(2) Chern density can be expressed in terms of the $\delta -$function $\delta (\phi) $. And one can find that the zero points of the vector fields $\phi$ are essential to the topological properties of a manifold. It is shown that there exists the crucial case of branch process at the zero points. Based on the implicit function theorem and the taylor expansion, the bifurcation of the Chern density is detailed in the neighborhoods of the bifurcation points of $\phi$. It is pointed out that, since the Chren density is a topological invariant, the sum topological chargers of the branches will remain constant during the bifurcation process.