The generation of the (k-1)-dimensional defect objects and their topological quantization

TL;DR

This paper investigates the topological structure and quantization of defects of arbitrary dimensions using $\phi$-mapping and topological current theory, revealing how topological quantum numbers are characterized and how defects originate from zero points of the mapping.

Contribution

It introduces a framework linking topological quantum numbers of defects to winding numbers, Hopf indices, and Brouwer degrees, and explains defect generation from zero points of the $\phi$-mapping.

Findings

Topological quantum numbers are described by winding numbers, Hopf indices, and Brouwer degrees.

All defects originate from zero points of the $\phi$-mapping.

The method applies to defects of arbitrary dimensions.

Abstract

In the light of $\phi $--mapping method and topological current theory, the topological structure and the topological quantization of arbitrary dimensional topological defects are investigated. It is pointed out that the topological quantum numbers of the defects are described by the Winding numbers of $\phi $--mapping which are determined in terms of the Hopf indices and the Brouwer degrees of $\phi$--mapping. Furthermore, it is shown that all the topological defects are generated from where $\vec \phi =0$, i.e. from the zero points of the $\phi $--mapping.