Classical and quantum Lagrangian field theories with boundaries

Abstract: Topological quantum field theories may be defined as appropriate functors from a cobordism category. The classical and perturbative versions thereof have started to be investigated only recently. We discuss topological field theories in the AKSZ formalism (this includes Chern-Simons and BF theories as well as the Poisson sigma model). In this case, to boundary components (objects) one can associate a symplectic manifold and a coisotropic submanifold thereof, whereas to cobordisms one associates canonical relations. This very natural construction (a special case with more structure than the general one recently introduced by V. Fock), yields to new viewpoints on, say, the moduli space of flat connections or the symplectic groupoid of a Poisson manifold. This also constitutes the starting point for the perturbative quantization of these theories. The possibility of including boundaries of boundaries (and so on) naturally yields to a description such that, eventually, one may hope to be able to reconstruct perturbative topological field theories by gluing simple pieces.