Stable Defects in Ordered Media

Abstract

Nature provides us with many examples of so-called “ordered media.” The best-known are liquid crystals (nematics, smectics, cholesterics, etc.), but a precise definition of such an ordered structure leads to some conceptual difficulties. Physicists are now so well acquainted with the notion of symmetries--and the corresponding mathematical notion of groups--that it is quite difficult for them to give up this idea in the case of objects which locally exhibit such a symmetry but in a variable manner, as in the case of a crystal in which the mesh of the lattice varies according to a global law of the entire medium. Mathematicians, on the other hand, have at their disposal a notion which at first glance seems to be very well suited to describing such objects, namely the notion of pseudo-group. Let us recall that a pseudo-group Г in a domain V of Euclidean space is defined by a family of local homeomorphisms h_ji having as source a neighborhood U_i in V, and as target an open set V_j in V. If the target V_j of h_ji is contained in the source U_j of h_kj where h_kj is another morphism of the pseudo-group Г, then the composed map h_kj o h_jialso belongs to the family Г. Identity homeomorphisms U_i ≃ U_i also belong to Г. Intuitively, a pseudo-group differs from a group by the fact that the composition of two maps of the pseudo-group may not be defined. As a result, pseudo-groups no longer have that rich mathematical structure associated with groups, which are so interesting to physicists (representations, invariants, etc.).