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

It is shown that the physical phase space of the γ-deformed Hamiltonian lattice in the 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. Here, L and V are, respectively, the total number of links and vertices of the lattice. The deformation parameter γ is identified with 2π/k, where k is an integer appearing in the Chern-Simons action.