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

Let S be a subset of n points on a compact connected oriented surface M of genus g, and let G be a compact semisimple Lie group. The space of isomorphism classes of flat G-connections on P:=M∖S with fixed conjugacy class of monodromy around each point of S will be denoted by R. It is known that R has a natural symplectic structure. We relate R with the space of geodesic (4g+n)-gons in G. A natural 2-form on the space of geodesic (4g+n)-gons is constructed using the Killing form on Lie(G). We establish an identity between the symplectic form on R and this 2-form on geodesic (4g+n)-gons in G.