1208.3578 (Alexander D. Popov)

Integrable vortex-type equations on the two-sphere    [PDF]

Alexander D. Popov

We consider 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 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. 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 the charge (g-1)N.

View original: http://arxiv.org/abs/1208.3578