Non-Abelian Vortices on Riemann Surfaces: an Integrable Case
Abstract
We consider U(n+1) Yang-Mills instantons on the space × S2, where 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 × S2 are equivalent to non-Abelian vortex equations on . Solutions to these equations are given by pairs (A,φ), where A is a gauge potential of the group U(n) and φ 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 × S2 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.