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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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