Abstract

Exact solutions to the self-dual Yang-Mills equations over Riemann surfaces of arbitrary genus are constructed. They are characterized by the conformal class of the Riemann surface. They correspond to U(1) instantonic solutions for an Abelian-Higgs system. A functional action of a genus g Riemann surface is constructed, with minimal points being the two-dimensional self-dual connections. The exact solutions may be interpreted as connecting initial and final nontrivial vacuum states of a conformal theory, in the sense of Segal, with a Feynman functor constructed from the functional integral of the action.