Geometric theory of topological defects: methodological developments and new trends

TL;DR

This paper reviews the differential geometry of topological defects in liquid crystals, explores their connections to cosmology and high-energy physics, and discusses defect engineering for transport properties.

Contribution

It provides a comprehensive overview of the geometric theory of liquid crystal defects and highlights interdisciplinary connections and recent advances in defect engineering.

Findings

Topological defects modify transport properties as if in non-Euclidean geometry.

Deep connections exist between liquid crystal defects and cosmology/high-energy physics.

Review of defect engineering techniques for controlling transport phenomena.

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.