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An analysis of the two-vortex case in the Chern-Simons Higgs model. (English) Zbl 0928.58021
The authors construct in 2+1 dimensional Minkowski space the Chern-Simons-Higgs model described by the Lagrangian density \[ {\mathcal L}(A)=D_\alpha \Phi {\overline D_\alpha \Phi}+\tfrac{k}{4} \varepsilon^{\alpha \beta \gamma} F_{\alpha \beta} A_{\gamma} -k^{-2} | \Phi | ^2 (1-| \Phi | ^2)^2 \] corresponding to a complex scalar field \(\Phi\) coupled to a gauge field \(A\) (\(F_{\alpha \beta} =\partial_\alpha A_\beta -\partial_\beta A_\alpha\), \(D_\alpha \Phi =\partial_\alpha \Phi -iA_\alpha \Phi\)). They investigate the stationary configurations of periodic vortices: these are solutions on some lattice in \({\mathbb R}^2\) which are gauge equivalent on the boundary of a fundamental domain \(\Omega\). The flux through \(\Omega\) is quantized: \(\int_\Omega F_{12} dx=2\pi N\), \(N\) being the vortex number.
The existence results hold for arbitrary \(N\). Some finer variational results and asymptotic results for coupling \(k \to 0\) are known only for \(N=1\). The authors present a variational existence proof of vortex solutions for \(N=2\) and analyze their asymptotic behaviour for \(k \to 0\). For \(N>2\), except in special cases, the situation is unclear.

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53Z05 Applications of differential geometry to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
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