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Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In particular I am interested in the case of a infinite cylinder $S^1 \times \mathbb{R}$.

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For finite-energy vortices on a finite-type Riemannian surface with cylindrical ends, there is still a non-negative integer parameter, the vortex number $N$, and the moduli space is still canonically diffeomorphic to the $N$th symmetric product by the map that takes a gauge-equivalence class of vortices $[A,\phi]$ to $\phi^{-1}(0)$. One can prove that such vortices extend over the puncture, whereupon the usual methods apply.

Some references:

1) The case of the complex plane was treated in the book "Vortices and monopoles" by Jaffe-Taubes.

2) The case of a cylinder is explicitly treated, by a different method, in a paper by Frauenfelder:

http://arxiv.org/abs/math/0507285

3) One can regard the vortex equations as dimensional reductions of the Seiberg-Witten equations. There is a comprehensive treatment of those equations in the presence of cylindrical ends in Kronheimer and Mrowka's book "Monopoles and 3-manifolds". They also discuss Atiyah-Patodi-Singer boundary conditions in the case where there is a boundary.

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