Integrable vortex-type equations on the two-sphere

TL;DR

This paper studies vortex-type equations derived from Yang-Mills instantons on the product of a sphere and a Riemann surface, showing integrability conditions and solutions relating to instanton charges.

Contribution

It introduces a reduction of Yang-Mills instanton equations to vortex equations on S^2 and demonstrates their integrability when scalar curvature vanishes, linking vortex solutions to instanton charges.

Findings

Vortex equations are integrable when scalar curvature vanishes.

Solutions exist for any topological charge N.

Vortex solutions correspond to Yang-Mills instantons with specific charges.

Abstract

We consider the Yang-Mills instanton equations on the four-dimensional manifold S^2xSigma, where Sigma is a compact Riemann surface of genus g>1 or its covering space H^2=SU(1,1)/U(1). Introducing a natural ansatz for the gauge potential, we reduce the instanton equations on S^2xSigma to vortex-type equations on the sphere S^2. It is shown that when the scalar curvature of the manifold S^2xSigma vanishes, the vortex-type equations are integrable, i.e. can be obtained as compatibility conditions of two linear equations (Lax pair) which are written down explicitly. Thus, the standard methods of integrable systems can be applied for constructing their solutions. However, even if the scalar curvature of S^2xSigma does not vanish, the vortex equations are well defined and have solutions for any values of the topological charge N. We show that any solution to the vortex equations on S^2 with a fixed topological charge N corresponds to a Yang-Mills instanton on S^2xSigma of charge (g-1)N.