On the introduction of a boundary in topological field theories

We study the consequences of the presence of a boundary in topological field theories in various dimensions. We characterize, univocally and on very general grounds, the field content and the symmetries of the actions which live on the boundary. We then show that these actions are covariant, despite appearances. We show also that physically relevant theories like the 2D Luttinger liquid model, or the 4D Maxwell theory, can be seen as boundary reductions of higher dimensional topological field theories, which do not display local observables.