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A question on the nature of the vortex number

In the Yang-Mills-Higgs (also called magnetic Ginzburg-Landau) model in the plane the energy has the form $$ E(A,\phi)=\int_{\mathbb{R}^2}\left(|(d-iA)\phi|^2+\frac{1}{2}F_{jk}F_{jk}+\frac{1}{4}(1-|\...