Liquid crystals are complex materials that share properties of both solids and liquids. This is a consequence of complex anisotropic molecules that permit establishing phases with orientational and positional orders. Thus, a large variety of phases and phase transitions can occur in these systems. After a detailed description of general features of these materials, the tensorial nature of the orientational order parameter is discussed. Then, the Landau–de Gennes theory is developed for the isotropic–nematic transition. Later, positional degrees of freedom are included to account for the nematic–smectic transition. Next, the theory is generalized to include fluctuations, distortions and the effect of an external field. In the last part, topological defects are discussed with a particular emphasis on defects such as skyrmions and merons which can form in chiral liquid crystals such as cholesteric and blue phases. Finally, the analogy of these classes of defects with those occurring in non-collinear magnetic materials is considered.

We stay in the framework of the low-energy effective theory of QCD in terms of Nambu–Goldstone bosons fields and consider effects due to their topology. We distinguish the cases of Nf = 2 or Nf >= 3 light quark flavors and discuss in both cases how the gauge anomaly cancelation is manifest in the effective theory, the role of G-parity, and the neutral pion decay into two photons, which does not explicitly depend on the number of colors, Nc . For Nf >= 3 we introduce the Wess–Zumino–Novikov–Witten term in a 5th dimension, we discuss the intrinsic parity of light meson fields and their electromagnetic interactions. In this context, we clarify the question whether there is low-energy evidence for Nc = 3, and we address again the role of technicolor.