Poisson Brackets and Flat Connections

A flat connection with regular singularities on a complex plane is characterized by its residues at singular points. A theorem of Hitchin establishes a surprizing link between the linear Kirillov-Kostant-Souriau (KKS)Poisson bracket on the residues and the Goldman bracket on traces of holonomies corresponding to closed loops. In this talk, we will show how to generalize this result to regularized holonomies corresponding to paths starting and ending at singular points. Our main technical tool is a version of the Drinfeld's pentagon equation suitable for paths with self-intersections.