Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

Abstract

We consider U(n+1) Yang-Mills instantons on the space Σ\times S^2, where Σis a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on Σ\times S^2 are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,ϕ), where A is a gauge potential of the group U(n) and ϕis a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g>1, when Σ\times S^2 becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.

8 pages; v2: typos fixed

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