Symplectic structure of the moduli space of flat connection on a Riemann surface (Q1894855)

The authors consider the canonical symplectic structure on the moduli space \(\mathcal M\) of flat \(\mathcal G\)-connections on a Riemann surface of genus \(g\) with \(n\) marked points. A combinatorial description of this symplectic structure is given and its efficient formula is obtained for the case of a surface with marked points. The relation of the symplectic structure on \(\mathcal M\) and Poisson-Lie symplectic structures are studied. It is shown that for \(\mathcal G\) being a semisimple Lie algebra, the symplectic form may be represented as a sum of \(n\) copies of the Kirillov symplectic form on the orbit of dressing transformations in the Poisson- Lie group \(G^*\) (corresponding to the Lie algebra \(\mathcal G\)) and \(g\) copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group \(G\).