Three lectures on derived symplectic geometry and topological field theories

Abstract

We give an informal introduction to the new field of derived symplectic geometry, and present some applications to topological field theories. We in particular try to explain that derived symplectic geometry provides a suitable framework for the so-called AKSZ construction (after Alexandrov–Kontsevich–Schwartz–Zaboronski). We start with a brief summary of the main features of derived algebraic geometry. We then continue with the definition of n-symplectic and Lagrangian structures, after Pantev–Toën–Vaquié–Vezzosi (PTVV), and provide examples such as •(for shifted symplectic structures) BG, [g∗/G], mapping stacks with symplectic target and “compact oriented” source.•(for Lagrangian structures) moment maps, quasi-Hamiltonian structures, mapping stacks with boundary conditions. We finally explain how this can be used to construct (fully extended) topological field theories with values in (higher) categories of Lagrangian correspondences.