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

It is shown that the physical phase space of \g-deformed Hamiltonian lattice Yang-Mills theory, which was recently proposed in refs.[1,2], 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 correspondingly a total number of links and vertices of the lattice. The deformation parameter \g is identified with $\frac {2\pi}{k}$ and $k$ is an integer entering the Chern-Simons action.Comment: 12 pages, latex, no figure