Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

TL;DR

This paper links non-Abelian vortex equations on Riemann surfaces to integrable systems by showing their equivalence to Lax pair compatibility conditions, enabling solution construction via integrable methods.

Contribution

It demonstrates that non-Abelian vortex equations on certain Riemann surfaces are integrable by deriving a Lax pair formulation, extending the understanding of their mathematical structure.

Findings

Vortex equations are equivalent to Lax pair compatibility conditions for genus g>1.

Solutions can be constructed using integrable systems techniques.

The model relates to gravitational instantons and non-Abelian Higgs theories.

Abstract

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.