The Wu-Yang potential of magnetic skyrmion from SU(2) flat connection

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 σ3. The components of the U (1) Wu-Yang curvature correspond to the emergent electromagnetic field of magnetic skyrmion.

Gauge Gravity Vacuum in Constraintless Clairaut-Type Formalism

“…are independent of the unresolved velocities v α since rank W AB = r. Additionally, the r.h.s. of (23) does not depend on the degenerate velocities v α ∂H phys ∂v α = 0, (25) which justifies the term "physical". The Hamilton-Clairaut system which describes any singular Lagrangian classical system (satisfying the second-order Lagrange equations) has the form…”

“…Bae, Cho and Kimm later clarified that this internal vector did not introduce two degrees of freedom requiring to be fixed but non-canonical DOFs without EOMs [18], while the proposed constraint was merely a consistency condition. The interested reader is referred to [6,[19][20][21] for further details (see, also [22,23]). Cho et al [24] approached the issue with Dirac quantisation using second-order restraints.…”

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