Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone

Damien Calaque, Pavel Safronov

Abstract

This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.

Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone

Abstract

This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.
Paper Structure (24 sections, 52 theorems, 157 equations)

This paper contains 24 sections, 52 theorems, 157 equations.

Table of Contents

  1. Preliminaries
  2. Derived algebraic geometry
  3. Groupoids in derived algebraic geometry
  4. Sheaves of $\mathcal{O}$-modules
  5. Total spaces
  6. Functoriality of the cotangent complex
  7. Shifted symplectic and Lagrangian structures
  8. Shifted presymplectic and isotropic structures
  9. Non-degeneracy of presymplectic structures
  10. Orientations
  11. AKSZ construction
  12. The $\infty$-category of shifted Lagrangian correspondences
  13. Shifted cotangent bundles
  14. Shifted cotangent bundles
  15. Functoriality of the cotangent bundle
  16. ...and 9 more sections

Key Result

Lemma 1.2

Let $f\colon X\rightarrow Y$ be a morphism of derived prestacks, $\mathcal{F}\in\mathbf{QCoh}(X)$ and $V\in\mathbf{Perf}(Y)$. Then the natural morphism is an isomorphism.

Theorems & Definitions (178)

  • Example 1.1
  • Lemma 1.2
  • proof
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Proposition 1.5
  • proof
  • Proposition 1.6
  • proof
  • ...and 168 more