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 . R2,1, or on a surface conformally related to a hyperbolic affine sphere in . R3. 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. © 2012 Elsevier B.V.
Dunajski, M. (2012). Abelian vortices from sinh-Gordon and Tzitzeica equations. Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 710(1), 236–239. https://doi.org/10.1016/j.physletb.2012.02.078