Physical phase space of the lattice Yang-Mills theory and moduli space of flat connections on a Riemann surface

It is shown that the physical phase space of γ-deformed, Hamiltonian-lattice Yang-Mills theory coincides as a Poisson manifold with the moduli space of flat connections on a Riemann surface with (L - V + 1) handles and, therefore, with the physical phase space of the corresponding (2 + 1) dimensional Chern-Simons model, where L and V are, respectively, the total number of links and vertices of the lattice. The deformation parameter γ is identified with 2 π/κ and κ is an integer entering the Chern-Simons action