Topological excitations

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Topological excitations are certain features of classical solutions of gauge field theories.

Namely, a gauge field theory on a manifold ${\displaystyle M}$ with a gauge group ${\displaystyle G}$ may possess classical solutions with a (quantized) topological invariant called topological charge. The term topological excitation especially refers to a situation when the topological charge is an integral of a localized quantity.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

Examples: [1]

1) ${\displaystyle M=R^{2}}$, ${\displaystyle G=U(1)}$, the topological charge is called magnetic flux.

In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B passing through that surface. The SI unit of magnetic flux is the weber (Wb), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the magnetic flux.

2) ${\displaystyle M=R^{3}}$, ${\displaystyle G=SO(3)/U(1)}$, the topological charge is called magnetic charge.

The concept of a topological excitation is almost synonymous with that of a topological defect.

A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological soliton occurs in old-fashioned coiled telephone handset cords, which are usually coiled clockwise. Years of picking up the handset can end up coiling parts of the cord in the opposite counterclockwise direction, and when this happens there will be a distinctive larger loop that separates the two directions of coiling. This odd looking transition loop, which is neither clockwise or counterclockwise, is an excellent example of a topological soliton. No matter how complex the context, anything that qualifies as a topological soliton must at some level exhibit this same simple issue of reconciliation seen in the twisted phone cord example.

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References

1. F. A. Bais, Topological excitations in gauge theories; An introduction from the physical point of view. Springer Lecture Notes in Mathematics, vol. 926 (1982)