# The Wu-Yang potential of Magnetic Skyrmion from SU(2) Flat Connection

## TL;DR

This paper explores the topological and gauge-theoretic origins of magnetic skyrmions using $SU(2)$ flat connections, linking their properties to Wu-Yang potentials and curvatures in gauge theory.

## Contribution

It introduces a novel gauge-theoretic framework expressing magnetic skyrmions' structure via $SU(2)$ flat connections and Wu-Yang potentials, providing new insights into their topological nature.

## Key findings

- Magnetic skyrmions' inner structure relates to $U(1)$ Wu-Yang curvature.
- Skyrmions can be generated through specific $SU(2)$ gauge transformations.
- The emergent electromagnetic field of skyrmions corresponds to components of Wu-Yang curvature.

## Abstract

The theoretical research of the origin of magnetic skyrmion is very interesting. By using decomposition theory of gauge potential and the gauge parallel condition of local bases of $su(2)$ Lie algebra, its $SU(2)$ gauge potential is expressed as flat connection. As an example of application, we obtain the inner topological structure of second Chern number by $SU(2)$ flat connection method. It's well known that if magnetic monopole exists in electrodynamics, its Wu-Yang potential is indispensable in $U(1)$ invariant electromagnetic field. In $2$-dim magnetic materials, we prove that if magnetic skyrmion exists, its integral kernel must be $U(1)$ Wu-Yang curvature, where its $U(1)$ Wu-Yang potential is the projection of $SU(2)$ flat connection on $su(2)$ local Cartan subalgebra. The magnetic skyrmion can be created by performing concrete $SU(2)$ local gauge transformation to $su(2)$ Cartan subalgebra $\sigma_3$. The components of the $U(1)$ Wu-Yang curvature correspond to the emergent electromagnetic field of magnetic skyrmion