Derived stacks in symplectic geometry

Damien Calaque (1, 2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IUF - Institut universitaire de France

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.

Domaines: Géométrie symplectique [math.SG]; Physique mathématique [math-ph]; Histoire et perspectives sur les mathématiques [math.HO]

Damien Calaque. Derived stacks in symplectic geometry. Mathieu Anel; Gabriel Catren. New Spaces in Physics: Formal and Conceptual Reflections, Cambridge University Press, pp.155--201, 2021, 9781108854399. ⟨10.1017/9781108854399.007⟩. ⟨hal-01721996⟩