Lagrangian structures on mapping stacks and semi-classical TFTs

We extend a recent result of Pantev-Toen-Vaquie-Vezzosi, who constructed shifted symplectic structures on derived mapping stacks having a Calabi-Yau source and a shifted symplectic target. Their construction gives a clear conceptual framework for the so-called AKSZ formalism. We extend the PTVV construction to derived mapping stacks with boundary conditions, which is required in most applications to quantum field theories (see e.g. the work of Cattaneo-Felder on the Poisson sigma model, and the recent work of Cattaneo-Mnev-Reshetikhin). We provide many examples of Lagrangian and symplectic structures that can be recovered in this way. We finally give an application to topological field theories (TFTs). We expect that our approach will help to rigorously constuct a 2 dimensional TFT introduced by Moore and Tachikawa. A subsequent paper will be devoted to the construction of fully extended TFTs (in the sense of Baez-Dolan and Lurie) from mapping stacks.