Topology on the lattice; 2d QCD with a theta term Original Research Article

We study two dimensional U(N) and SU(N) gauge theories with a topological term on arbitrary surfaces. Starting from a lattice formulation we derive the continuum limit of the action which turns out to be a generalisation of the heat kernel in the presence of a topological term. In the continuum limit we can reconstruct the topological information encoded in the theta term. In the topologically trivial cases the theta term gives only a trivial shift to the ground state energy but in the topologically non-trivial ones it survives to be coupled to the dynamics in the continuum. In particular for the U(N) gauge group on orientable surfaces it gives rise to a phase transition at θ = π, similar to the ones observed in other models. Using the equivalence of 2d QCD and a 1d fermion gas on a circle we rewrite our result in the fermionic language and show that the theta term can be also interpreted as an external magnetic field imposed on the fermions.