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.