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Chapter 18 describes a few systems where the classical action has an infinite number of degenerate minima but, in the quantum theory, this degeneracy is lifted by barrier penetration effects. The simplest example is the cosine periodic potential and leads to the band structure. Technically, this corresponds to the existence of instantons, solutions to classical equations in imaginary time. In all examples, we show that the classical solutions are constrained by Bogomolnyi’s inequalities, which involve topological charges associated to a winding number and defining homotopy classes. In the case of quantum chromodynamics, this leads to the famous strong CP violation problem.

Chapter 13 is devoted to some aspects of quantum chromodynamics (QCD), the part of the Standard Model of particle physics responsible for strong interactions and based on an SU(3) gauge symmetry (the colour group) and gluon gauge fields. First, the geometry of non–Abelian gauge theories, based on parallel transport, is recalled. This leads naturally to the construction of lattice gauge theories with link variables and a plaquette action. The lattice model gives a hint of confinement. QCD is quantized in the temporal of Weyl gauge. Its renormalization involves the BRST symmetry. Its renormalization group properties with asymptotic freedom are emphasized. The infinite degeneracy of the semi–classical ground state can be associated to a winding number. Barrier penetration effects, related to the existence of instantons, lead to the existence of theta vacua and the problem of strong CP violation. Other issues considered are chiral symmetry and axial anomaly.

An important application of spin dynamics is the response of a magnetic material subjected to an external stimuli. In the previous chapter we discussed the response of primarily ferromagnets to temperature fluctuations that manifest itself to spin excitations and magnons. In this chapter, we are concerned about magnetic materials with more complicated magnetic texture, such as spin spirals and topological magnetic structures, in particular magnetic skyrmions. Magnetic skyrmions has many appealing and intriguing features that make them interesting both for possible applications but also from a pure theoretical point of view.

The standard model of fundamental interactions. A brief summary of the phenomenology of weak interactions. The construction of the electroweak theory and its experimental consequences. The deep inelastic scattering data as a motivation for quantum chromodynamics. Asymptotic freedom and the parton model. Quantum chromodynamics formulated on a space–time lattice. Non-trivial gauge field configurations and instantons. The meaning of the winding number. The strong CP problem and axions.