Shifted coisotropic structures for differentiable stacks

Abstract

We introduce a notion of coisotropics on 1-shifted symplectic Lie groupoids (i.e. quasi-symplectic groupoids) using twisted Dirac structures and show that it satisfies properties analogous to the corresponding derived-algebraic notion in shifted Poisson geometry. In particular, intersections of 1-coisotropics are 0-shifted Poisson. We also show that 1-shifted coisotropic structures transfer through Morita equivalences, giving a well-defined notion for differentiable stacks. Most results are formulated with clean-intersection conditions weaker than transversality while avoiding derived geometry. Examples of 1-coisotropics that are not necessarily Lagrangians include Hamiltonian actions of quasi-symplectic groupoids on Dirac manifolds, and this recovers several generalizations of Marsden-Weinstein-Meyer's symplectic reduction via intersection and Morita transfer.

48 pages; Final version

References in corpus (29)

• Relative formality theorem and quantisation of coisotropic submanifolds
• Shifted Poisson Structures and Deformation Quantization
• Vector bundles over Lie groupoids and algebroids
• Multiple Derived Lagrangian Intersections
• Lagrangian structures on mapping stacks and semi-classical TFTs
• Lie Groupoids and Differentiable Stacks
• On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds
• Perturbative algebraic quantum field theory
• Quasi-Hamiltonian reduction via classical Chern-Simons theory
• Non Abelian Differentiable Gerbes
• Derived coisotropic structures II: stacks and quantization
• Derived coisotropic structures I: affine case
• Lie 2-algebras of vector fields
• Riemannian metrics on differentiable stacks
• Shifted Poisson and symplectic structures on derived N-stacks
• The Normal Form Theorem around Poisson Transversals
• Shifted Coisotropic Correspondences
• Morita equivalences of vector bundles
• Shifted derived Poisson manifolds associated with Lie pairs
• Symplectic implosion and the Grothendieck-Springer resolution
• Stacky Hamiltonian actions and symplectic reduction
• Stacky Lie groups
• The volume of a differentiable stack
• Measures on differentiable stacks
• On the homotopy theory for Lie $\infty$-groupoids, with an application to integrating $L_{\infty }$-algebras
• Shifted symplectic reduction of derived critical loci
• On the coadjoint Virasoro action
• Discrete dynamics and differentiable stacks
• The van Est Map on Geometric Stacks