Flat connections on a punctured sphere and geodesic polygons in a Lie group

Abstract

Let G be a compact connected Lie group and X denote the complement of n distinct points of the sphere S². The space of isomorphism classes of flat G connections on X with fixed conjugacy class of holonomy around each of n punctures has a natural symplectic structure. This space is related to the space of geodesic n-gons in G. The space of geodesic polygons in G has a natural 2-form. It is shown that this 2-form coincides with symplectic form on the space of isomorphism classes of flat G-connections on X satisfying holonomy condition at the punctures.