On the one hand, symplectic geometry is a natural setting for classical mechanics: phase spaces appear to be symplectic manifolds (or variations of these). Indeed, a symplectic (or, rather, Poisson) structure is precisely what is needed to associate a dynamical system on the given space to each energy function. On the other hand, derived algebraic geometry has been invented in order to deal in a satisfying way with spaces that are singular, which is the case for many spaces appearing in algebraic geometry (moduli spaces) and in classical physics (spaces of solutions of equations of motion).
The theory of symplectic structures on derived stacks gave birth to what is now called derived symplectic geometry. Since then we have seen a rapid evolution of knowledge, and several applications as well (such as general existence theorems for symplectic structures and symmetric obstruction theories on moduli spaces, for instance). This project aims at providing new theoretical developments (shifted symplectic groupoids, for example) and new applications of derived symplectic geometry (to classical field theories, for instance).
On the foundational side (Main Goal A), our aim was to introduce the new notion of shifted symplectic groupoids and prove that they provide an alternative approach to shifted Poisson structures. Along the way, we intended be able to prove several conjectures that have recently been formulated by the PI and other people.
Applications are related to mathematical physics: interpretation of the Batalin–Vilkovisky formalism in terms of derived symplectic reduction (Main Goal C), constuction of fully extended topological field theories (Main Goal B). Quantization problems will also be discussed at the end of the proposal.
This project proposal lies at the crossroads of algebraic geometry, mathematical physics (in its algebraic and geometric aspects) and higher algebra.
Conclusion for the Main Goal A: as intended, we have developped the theory of shifted symplectic groupoids, explained its relation to lagrangian morphisms, and applied this to the construction of a lagrangian version of the deformation to normal cone.
Conclusion for the Main Goal B: as intended, we constructed a fully extended topological field theory with values in a higher symmetric monoidal category of iterated lagrangian correspondences. There are also other results about field theories that have been obtained (both of general nature and for specific theories).
Conclusion for Main Goal C : as intended, we developped the theory of derived symplectic reduction, and explained how it explains some aspects of the classical BV formalism.
A few results on quantization of some very specific constructions of topological field theories have been ontained.
