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Shifted symplectic rigidification

Hyeonjun Park, Jemin You

Abstract

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any $B\mathbb{G}_m$-action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.

Shifted symplectic rigidification

Abstract

We construct shifted symplectic derived enhancements on rigidified moduli spaces of sheaves on Calabi-Yau varieties of dimension at least two. More generally, we prove that any -action on a non-positively-shifted symplectic derived Artin stack is Hamiltonian. We provide a symplectic rigidification functor as the left adjoint to the trivial action functor in symplectic categories with Lagrangian correspondences. We also descend the Lagrangian correspondence of short exact sequences of sheaves to rigidified moduli spaces.

Paper Structure

This paper contains 27 sections, 40 theorems, 214 equations.

Table of Contents

  1. Symplectic categories
  2. Differential forms
  3. Symplectic categories
  4. Lagrangian categories
  5. Adjunction
  6. Lagrangian fibrations
  7. Cotangent bundles
  8. Hamiltonian reduction
  9. Group actions
  10. Symplectic actions
  11. Hamiltonian actions
  12. Change of groups
  13. Symplectic rigidification
  14. Results
  15. Invariance
  16. ...and 12 more sections

Key Result

Theorem A

Let $M$ be a $d$-shifted symplectic derived Artin stack. If $d \leq 0$, then any $B\mathbb{G}_m$-action on $M$ is Hamiltonian. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (97)

  • Theorem A: \ref{['Prop:BT-symp']}Prop:BT-symp, \ref{['Thm:Ham=Symp']}Thm:Ham=Symp
  • Corollary A
  • Theorem B: \ref{['Cor:rigid-adjunction']}Cor:rigid-adjunction
  • Corollary B: \ref{['Prop:RidigifiedExtCorr']}Prop:RidigifiedExtCorr
  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • proof : Proof of \ref{['Prop:LagComp-Functoriality']}Prop:LagComp-Functoriality
  • Proposition 2
  • ...and 87 more