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arXiv · 1201.0105

Abelian vortices from Sinh--Gordon and Tzitzeica equations

Maciej Dunajski
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Abstract

It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlevé transcendents are used to compute explicit values for the strength of the vortices.

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Maciej Dunajski. 2012-02-25. Abelian vortices from Sinh--Gordon and Tzitzeica equations. https://doi.org/10.1016/j.physletb.2012.02.078

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