Uniqueness of selfdual periodic Chern-Simons vortices of topological-type
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 the divisor \(Z(\phi)\); the set of vortex points of \(\phi\) in \(\mathbb{T}^2\), each repeated according to their multiplicity, is called the vortex number. Since \[ {2\over\kappa^2} \int_\Omega |\phi|^2(1- |\phi|^2)= 2\pi N, \] asymptotic behavior of \(\phi\) as \(\kappa\to 0\) is either \(|\phi|\to 1\) (topological type), or \(|\phi|\to 0\) (non-topological type) on \(\Omega\). It is known there exists the maximal solution \((A^{(\kappa)}_{\max},\phi^{(\kappa)}_{\max})_\pm\), \(\kappa\in (0,\kappa_C)\), such that \[ |\phi(x)|< |\phi^{(\kappa)}_{\max}(x)|, \quad x\in\mathbb{T}^2,\quad (F^{\max}_{1,2})_\pm\to \pm 2\pi \sum^N_{i=1} \delta_{z_j}, \] among the toplogical solutions of the selfdual equation for the given \(Z(\phi)= \{z_1,\dots, z_N\}\) (Th.2.1.). The main result of this paper is the uniqueness of topological solution of the selfdual equation if \(|\kappa|\) is small: That is, for sufficiently small \(\kappa\), any topological solution of the selfdual equation is shown to coincide with the maximal solution (Th.2.3). To set \(u= \log|\phi|^2\), the selfdual equation for the given \(Z(\phi)\) is transformed to the equation \[ -\Delta u={4\over \kappa^2} e^u(1- e^u)- 4\pi \sum^N_{j=1} \delta_{z_j}. \] The proof of Th.2.3 is reduced to the uniqueness problem for the solutions of the tranformed equation (Th.3.6), which follows from the existence of a local minimum of the functional \[ J_\lambda(\nu)= \int_{\mathbb{T}} |\nabla\nu|^2\, d\sigma+ {\lambda\over 2}\int_{\mathbb{T}^2} (e^u- 1)^2\, d\sigma+ {4\pi N\over |\mathbb{T}^2|} \int_{\mathbb{T}^2} \nu, \] \[ \nu= u- 4\pi \sum^N_{j=1} G(x, z_j),\quad -\Delta_x G(x,z)= \delta_z- {1\over|\mathbb{T}^2|},\quad \int_{\mathbb{T}^2} G(x,z)\,d\sigma_x= 0, \] (Th.3.7). This is proved in \S4, adopting a blow up analysis around the vortex point and radial symmetry of solutions of the transformed equation together with standard techniques in the study of elliptic equations. In the final remark (\S5), Th.3.7 is generalized when the location of vortex points varies depending on \(\kappa\) (Th.5.7).
scientific article; zbMATH DE number 5146202
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