Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

Details Non-Abelian Vortices on Riemann Surfaces: an Integrable Case Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

2008-01-05

arxiv.org/abs/0801.0808v2

Lett.Math.Phys.84:139-148,2008

Physics

High Energy Physics

High Energy Physics - Theory

8 pages; v2: typos fixed

Scientific paper

10.1007/s11005-008-0243-x

We consider U(n+1) Yang-Mills instantons on the space \Sigma\times S^2, where \Sigma is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n+1) instanton equations on \Sigma\times S^2 are equivalent to non-Abelian vortex equations on \Sigma. Solutions to these equations are given by pairs (A,\phi), where A is a gauge potential of the group U(n) and \phi is 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 \Sigma\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.

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