Chern-Simons action and disclinations
type: Item
instance of: scholarly article
title: Chern-Simons action and disclinations
zbMATH Open document ID: 1403.37079
DOI: 10.1134/S0081543818040107
author: M. O. Katanaev
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2018-10-23T00:00:00Z
review text: From the text: ``Most physical properties of crystals, such as plasticity, melting, growth, etc., are described by defects in the crystalline structure. One of the most promising approaches to describe defects is based on the Riemann-Cartan geometry with nontrivial curvature and torsion. In this approach, a crystal is considered as a manifold (continuous medium) with given unit vector field (spin structure). If dislocations are absent, then there exists a continuous displacement vector field which corresponds to diffeomorphisms of Euclidean space. When the displacement vector field has discontinuities, we say that there are defects, called dislocations. This leads to nontrivial geometry. Namely, dislocations correspond to nontrivial torsion, which has the physical meaning of the surface density of the Burgers vector. Defects (discontinuities) of the unit vector field are called disclinations. They correspond to a nontrivial curvature tensor, which has the physical meaning of the surface density of the Frank vector.'' In this paper the three-dimensional Chern-Simons action is considered in the framework of the geometric theory of defects. In Section 2 the general construction of the models based on the Chern-Simons action is given. In Section 3 the Hilbert-Einstein action is presented. In Section 4 the gauge fields for the inhomogeneous rotation group are considered. The relation between the Chern-Simons and Hilbert-Einstein actions is pointed out. In Section 5 a brief description of disclinations in the geometric theory of defects is given. In Section 6 the static distribution of linear disclinations using the Chern-Simons action is described. Particularly, the case of one straight disclination is analyzed. Finally, concluding remarks are given. One of them states that in this paper ``the first example of disclination described in the framework of the geometric theory of defects'' is given.
reviewed by: Cristian Lăzureanu
Mathematics Subject Classification ID: 37K25
Mathematics Subject Classification ID: 70G45
Mathematics Subject Classification ID: 70S10
Mathematics Subject Classification ID: 70G65
zbMATH DE Number: 6962310
zbMATH Keywords: Chern-Simons action
zbMATH Keywords: Hilbert-Einstein action
zbMATH Keywords: geometric theory of defects
zbMATH Keywords: disclinations
MaRDI profile type: Publication
Wikidata QID: Q125598566
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full work available at URL: s0081543818040107
OpenAlex ID: W2886722390
description: scientific article; zbMATH DE number 6962310