Integrability of Vortex Equations on Riemann Surfaces

Abstract

The Abelian Higgs model on a compact Riemann surface Σof genus g is considered. We show that for g > 1 the Bogomolny equations for multi-vortices at critical coupling can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. These vortices correspond precisely to SO(3)-symmetric Yang-Mills instantons on the (conformal) gravitational instanton Σ\times S^2 with a scalar-flat Kahler metric. Thus, the standard methods of constructing solutions and studying their properties by using Lax pairs (twistor approach, dressing method etc.) can be applied to the vortex equations on Σ. In the twistor description, solutions of the integrable vortex equations correspond to rank-2 holomorphic vector bundles over the complex 3-dimensional twistor space of Σ\times S^2. We show that in the general (nonintegrable) case there is a bijection between the moduli spaces of solutions to vortex equations on Σand of pseudo-holomorphic bundles over the almost complex twistor space.

16 pages; v2: typos fixed, clarifying comments added, published version

References in corpus (3)

• Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories
• TASI Lectures on Solitons
• Sigma Models with N=8 Supersymmetries in 2+1 and 1+1 Dimensions

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• Non-Abelian Vortices on Riemann Surfaces: an Integrable Case
• Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices
• Explicit Non-Abelian Monopoles and Instantons in SU(N) Pure Yang-Mills Theory
• Non Abelian Vortices as Instantons on Noncommutative Discrete Space