Topology on the lattice; 2d Yang-Mills theories with a $\vartheta$ term

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 nontrivial ones it remains 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 \theta= \pi, 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