Uniqueness of selfdual periodic Chern-Simons vortices of topological-type
type: Item
instance of: scholarly article
title: Uniqueness of selfdual periodic Chern-Simons vortices of topological-type
zbMATH Open document ID: 1123.35050
published in: Calculus of Variations and Partial Differential Equations
publication date: 2007-04-26T00:00:00Z
review text: Periodic topological type selfdual vortex solutions for the Chern-Simons model are shown to be uniquely determined by the location of their vortex points, when the Chern-Simons coupling parameter is small. The author remarks that the uniqueness result of planar topological selfdual Chern-Simons vortices in \(\mathbb{R}^2\), similar in spirit of this paper, is established by \textit{K. Choe} [J. Math. Phys. 46, 012305, 21 p. (2005; Zbl 1076.58012)]. Let \(\mathbb{R}^{1+2}\) be the Minkowski space with the metric \((x^0)^2- (x^1)^2- (x^2)^2\), \(A= -iA_\alpha dx^\alpha\) the potential field and \(\phi\) the Higgs scalar. Then the Lagrangean density of the Chern-Simons model is \[ {\mathcal L}_{CS}(A, \phi)=- {\kappa\over 4}\varepsilon^{\alpha\beta\gamma} A_\alpha F_{\alpha,\beta}+ D_\alpha\phi\overline{D^\alpha\phi}- {1\over\kappa^2} |\phi|^2(1- |\phi|^2)^2, \] where \(\varepsilon^{\alpha\beta\gamma}\) is the symmetric Levi tensor fixed by \(\varepsilon^{012}= 1\), \(D_\alpha= \partial_\alpha= \partial_\alpha- iA_\alpha\) and \(\kappa> 0\) is the Chern-Simons parameter. Then a static vortex configulation carrying minimal energy is realized by a solution \((A,\phi)\) of the following selfdual equation \[ D_\pm\phi= 0,\quad F_{12}= \pm{2\over\kappa^2} |\phi|^2(1-|\phi|^2),\quad 2A_0|\phi|^2= \kappa F_{12}, \] where \(D_\pm= D_1\pm iD_2\). By this first equation, \(\phi\) admits only isolated zeros with integral multiplicity. A zero of \(\phi\) is said to be a vortex point. The selfdual equation is invariant under the gauge tansformation \[ \phi\to e^{i\omega}\phi\quad A_j\to A_j+ \partial_j\omega,\quad j= 1,2,\quad A_0\to A_0. \] The gauge invariant periodic boundary condition is \[ \phi(x+ \vec a^k)= e^{i\omega_k(x)} \phi(x),\quad A_j(x+ \vec a^k)= A_j(x)+ \partial_j\omega_k(x),\quad j= 1,2, \] and \(A_0(x+ \vec a^k)= A_0(x)\). Let \(\mathbb{T}^2= \mathbb{R}^2/\vec a^1\mathbb{Z}\times \vec a^2\mathbb{Z}\) and \(\Omega\) its fundamental domain. Then the degree \(N\) of …
reviewed by: Akira Asada
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 58E50
zbMATH DE Number: 5146202
zbMATH Keywords: Chern-Simons model
zbMATH Keywords: self-dual equation
zbMATH Keywords: vortex number
zbMATH Keywords: 't Hooft boundary condition
author: G. Tarantello
MaRDI profile type: Publication
full work available at URL: s00526-006-0062-9
OpenAlex ID: W2063794666
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DOI: 10.1007/S00526-006-0062-9
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description: scientific article; zbMATH DE number 5146202