Abelian vortices from Sinh--Gordon and Tzitzeica equations

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 ${\mathbb{R}}^{2,1}$, or on a surface conformally related to a hyperbolic affine sphere in ${\mathbb{R}}^3$. 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.

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

References in corpus (2)

• Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II
• Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III

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• Integrable Abelian Vortex-like solitons
• Instanton Solutions from Abelian Sinh-Gordon and Tzitzeica Vortices
• Vortex Motion on Surfaces of Small Curvature
• Nineteen vortex equations and integrability
• Skyrmions from gravitational instantons
• The volume of a vortex and the Bradlow bound
• Vortices over Riemann surfaces and dominated splittings
• Elizabethan vortices
• Polynomial cubic differentials and convex polygons in the projective plane
• Self-dual gravity via Hitchin's equations
• Non-abelian vortices on CP^1 and Grassmannians
• Polymer plats and multicomponent anyon gases