We prove Derrick's theorem about scalar field solitons, then we derive the Bogomolnyi bound for the energy of scalar field configurations in 1+1 dimensions and consider the example of the Higgs system and its kink soliton. Then we consider spontaneous symmetry breaking in the Abelian–Higgs syste, and the different fluctuations and their masses, as well as the description of this system in an unitary gauge. We end with a quick treatment of the non-Abelian Higgs system.

In this chapter, we consider the vortex of the Abelian–Higgs system, the Nielsen–Olesen vortex (or ANO vortex). We find the Bogomolnyi bound for the energy of the system in terms of a topological charge. For a certain relation between coupling, known as a BPS limit, we find that the bound is saturated by a configuration with topological charge, i.e., magnetic charge. In this limit, we find BPS equations, which are solved by a vortex ansatz, for a vortex solution.The properties of the solutionand its application to superconductivity are explored.

We study the 't Hooft-Polyakov monopole solution of the nonabelian Georgi-Glashow model, a model with the gauge group SU(2)=SO(3) and scalar fields in the 3 representation. After setting up the model, and finding the vacuum manifold, we solve for the monopole through an ansatz. We then study the topology of the solution through an analysis of homotopy groups. We derive a Bogomolnyi bound, and a BPS limit (for the scalar quartic coupling to vanish), in which we get linear BPS equations, which are solved exactly. The topology of the BPS monopole is compared with the topology of the Dirac monopole, and an embedding relation of the latter in the former is found.