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 (18)

• Perturbative Gauge Theory As A String Theory In Twistor Space
• Solitons in the Higgs phase -- the moduli matrix approach --
• Supersymmetric Solitons and How They Help Us Understand Non-Abelian Gauge Theories
• An Alternative String Theory in Twistor Space for N=4 Super-Yang-Mills
• Conformal Supergravity in Twistor-String Theory
• TASI Lectures on Solitons
• The N=2(4) string is self-dual N=4 Yang-Mills
• Volume of Vortex Moduli Spaces
• Dimensional reduction, SL(2,C)-equivariant bundles and stable holomorphic chains
• Quiver Gauge Theory of Nonabelian Vortices and Noncommutative Instantons in Higher Dimensions
• The Topological B-model on a Mini-Supertwistor Space and Supersymmetric Bogomolny Monopole Equations
• Existence and Deformation Theory for Scalar-Flat Kaehler Metrics on Compact Complex Surfaces
• Rank Two Quiver Gauge Theory, Graded Connections and Noncommutative Vortices
• Thermodynamics of Vortices in the Plane
• The Magnetic Monopole Seventy-Five Years Later
• The dynamics of vortices on S^2 near the Bradlow limit
• Sigma Models with N=8 Supersymmetries in 2+1 and 1+1 Dimensions
• On Explicit Point Multi-Monopoles in SU(2) Gauge Theory

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