Integrability of Vortex Equations on Riemann Surfaces

The Abelian Higgs model on a compact Riemann surface \Sigma of 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 \Sigma\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 \Sigma. 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 \Sigma\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 \Sigma and of pseudo-holomorphic bundles over the almost complex twistor space.