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Vortices on the cylinder. (English) Zbl 1119.58010

The author studies the vortex equations which describe the absolute minima of the Yang-Mills-Higgs functional. Given a Riemann surface \(\Sigma\) with a unitary line bundle \(L\), this functional is defined on the space formed by all unitary connections on \(L\) and smooth sections of \(L\). By using the finite-dimensional approximation technique of Furuta, Kronheimer, and Manolescu, the author reproves the result of Jaffe and Taubes that for a given vortex number \(N\) the moduli space of the solutions to the vortex equations on the cylinder \(Z\) modulo gauge equivalence is the \(N\)-fold symmetric power of \(Z\).

MSC:

58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
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