The distinction between "ordinary" and topological charges comes from the fact that the conservation of the ordinary charges is a consequence of the Noether's theorem, i.e., when the system under consideration possesses a symmetry, then according to the Noether's theorem, the corresponding charge is conserved.

Topological charges, on the other hand, do not correspond to a symmetry of the given system model, and they stem from a procedure that can be called topological quantization. Please see the seminal work by Orlando Alvarez explaining some aspects of this subject. These topological charges correspond to topological invariants of manifolds related to the physical problem.

One of the most basic examples is the Dirac's quantization condition, which implies the quantization of the magnetic charge in units of the reciprocal of the electric charge. This condition is related to the quantization of the first Chern class of the quantum line bundle. It is also possible to obtain the quantization condition from single-valuedness requirement of the path integral. The existence of the topological invariants is related to a nontrivial topology of the manifold under consideration, for example nonvanishing homotopy groups, please, see the following review by V.P. Nair.

Of course, topological charges can also be non-Abelian; a basic example of this phenomenon, is the 't Hooft-Polyakov monopole, where these solutions have non-Abelian charges corresponding to weight vectors of the dual of the unbroken gauge group. Please see the following review by Goddard and Olive.

It should be emphasized that the distinction between ordinary charges and topological charges is model dependent, and "ordinary" charges in some model of a system emerge as topological charges in another model of the same system. For example the electric charge of a particle can be obtained as a topological charge in a Kaluza-Klein description. Please see section 7.6 here in Marsden and Ratiu.

Topological charges correspond sometimes to integer parameters of the model, for example, Witten was able to obtain the quantization of the number of colors from the (semiclassical) topological quantization of the coefficient of the Wess-Zumino term of the Skyrme model.

A simple example, where quantum numbers can be obtained as topological charges is the isotropic harmonic oscillator. If we consider an isotropic Harmonic oscillator in two dimensions then its energy hypersurfaces are $3$-spheres, which can be viewed as circle bundles over a $2$-sphere by the Hopf fibration. The $2$-spheres are the reduced phase spaces of the (energy hypersurfaces) of the two dimensional oscillator. In the quantum theory, the areas of these spheres need to be quantized, in order to admit a quantum line bundle. This quantization condition is equivalent to the quantization of the energy of the harmonic oscillator.

Actually, these alternative representations of physical systems, such that ordinary charges emerge as topological charges offer possible explanations for the quantization of these charges in nature (the Kaluza-Klein model for the electric charge, for example).

A current direction of research along to these lines is to find topological "explanations" to fractional charges. One of the known examples of is the expanation of the fractional hypercharge of the quarks (in units of $\frac{1}{3}$), which can be explained from the requirement of the anomaly cancelation (which is topological) of the standard model, where the contribution of the quarks must be multiplied by $3$ (due to the three colors). In addition to anomalies, it is known that the existence of fields of different irreducible representations in the same model and separately knotted configurations may give rise to fractional charges.

This post imported from StackExchange Physics at 2014-04-04 16:17 (UCT), posted by SE-user David Bar Moshe