Geometric theory of topological defects: methodological developments and new trends

Abstract

Liquid crystals generally support orientational singularities of the director field known as topological defects. These latter modifiy transport properties in their vicinity as if the geometry was non-Euclidean. We present a state of the art of the differential geometry of nematic liquid crystals, with a special emphasis on linear defects. We then discuss unexpected but deep connections with cosmology and high-energy-physics, and conclude with a review on defect engineering for transport phenomena.

Combined Screw and Wedge Dislocations

“…However, metric (41) is defined for all ν and consequently satisfies vacuum Einstein equations. It is the analytic continuation of the solution for all ν ∈ R.…”

“…Last year, great attention was paid to analogue gravity [39], which investigates analogues of gravitational field phenomena in condensed matter physics. In particular, the behavior of fields in curved spacetime can be studied in the laboratory (see [40,41] for review). In analogue gravity, nontrivial metrics arise as the consequence of field equations for condensed matter, which differ from gravity equations.…”

Global symmetry breaking and topological defects

“…Another important aspect to take into account in the process of spontaneous symmetry breaking are the possible topological defects generated by the breaking. Topological defects [30] [31] [32] are concentrations of energy that are generated after the spontaneous breaking of a symmetry, in case the resulting vacuum has a non-trivial topology. To understand non-trivial topology, let's look at a couple of examples: domain walls [33] [34] and cosmic strings [35] [36].…”

String theory and theory of everything: Review research

“…Among the latest studies of string theory, it is worth highlighting the studies of J. Gomis [5] (on the study of the limitations of bosonic string theory), S.P. De Alwis [6] (on the combination of string theory with Wilsonian effective field theory), S. Raucci [7] (on studies of non-super-symmetric strings, in particular, their vacuum states), S. Brahma [8] (on studies of the matrix model of the cosmogony of the Universe), S. Fumeron [9] (on studies of the topology of compactification spaces of additional dimensions of string theory), etc. Many studies and layers of hypotheses require systematisation and building a chronological vertical containing ordered and mutually consistent concepts, which in the future becomes a potential support structure for the development of the investigated sector of physics.…”