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

Abstract

It is shown that the physical phase space of -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 is identified with 2/k and k is an integer entering the Chern-Simons action.