Spontaneous Breaking of Euclidean Invariance and Classification of Topologically Stable Defects and Configurations of Crystals and Liquid Crystals

We show how many mesomorphic states illustrate the following general scheme: The symmetry group of an equilibrium state of Euclidean-invariant quantum statistical mechanics is a subgroup $H$ of the Euclidean group $E$ such that the orbit $\frac{E}{H}$ is compact. Moreover, the homotopy groups of $\frac{E}{H}$ yield a classification of the topologically stable defects and configurations of these ordered media. This suggests a predictive value of this scheme for yet unobserved media and for defects.