Abelian vortices from Sinh--Gordon and Tzitzeica equations

Details Abelian vortices from Sinh--Gordon and Tzitzeica equations Abelian vortices from Sinh--Gordon and Tzitzeica equations

2011-12-30

arxiv.org/abs/1201.0105v2

Physics

High Energy Physics

High Energy Physics - Theory

10 pages. Possible physical applications discussed + one reference added. Final version, to appear in Physics Letters B

Scientific paper

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 Painlev\'e transcendents are used to compute explicit values for the strength of the vortices.

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