Derived stacks in symplectic geometry

TL;DR

This survey explores how derived stacks in symplectic geometry provide a unified framework for understanding classical topological field theories, with examples like Chern-Simons theory and moduli spaces.

Contribution

It introduces derived symplectic geometry as a tool to connect various perspectives on classical Chern-Simons theory and moduli spaces of flat connections.

Findings

Derived stacks unify different viewpoints in symplectic geometry.

Derived symplectic geometry clarifies structures in topological field theories.

Examples include moduli spaces of flat G-bundles and local systems.

Abstract

This is a survey paper on derived symplectic geometry, that will appear as a chapter contribution to the book "New Spaces for Mathematics and Physics", edited by Mathieu Anel and Gabriel Catren. Our goal is to explain how derived stacks can be useful for ordinary symplectic geometry, with an emphasis on examples coming from classical topological field theories. More precisely, we use classical Chern-Simons theory and moduli spaces of flat $G$-bundles and $G$-local systems as leading examples in our journey. We start in the introduction by reviewing various point-of-views on classical Chern--Simons theory and moduli of flat connections. In the main body of the Chapter we try to convince the reader how derived symplectic geometry (after Pantev-To\"en-Vaqui\'e-Vezzosi somehow reconciles all these different point-of-views.