The poisson structure on the moduli space of flat connections and chord diagrams

Abstract

We Introduce the notion of chord diagrams on arbitrary compact (possibly punctured) oriented surfaces. In the case of the 2-spheres these are just the usual chord diagrams used in test study of Vassiliev invariants of links. We consider the algebra of chord diagrams on a surface and prove that this algebra has a natural Poisson structure. Suppose now that G is a Lie group with an invariant bilinear form on g = Lie( G). We can associate to each chord diagram (coloured by representations of G) a function on the moduli space of flat G-connections on the surface. Our main result states that this map is a Poisson algebra homomorphism. Moreover, for most classical groups we prove that any algebraic function on moduli space can be obtained this way and we conjecture that this holds for all simple groups. In this way we obtain a universal description of the Poisson algebra of the moduli space, decoupling the Lie group in question.